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Calculus exercises: part I

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1 Examples

Example. This is a different kind of example. Evaluate: $$\lim_{x \to 5} \frac{2^{x} - 32}{x - 5}. $$

It's just a limit. But we recognize that this is the derivative of some function. We compare the expression to the formula in the definition: $$ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}, $$ and match. So, we have here: $$a = 5 ,\ f(x) = 2^{x}, \ f(5) = 2^{5} = 32.$$ Therefore, our limit is equal to $f'(5)$ for $f(x) = 2^{x}$. Compute: $$f'(x) = (2^{x})' = 2^{x} \ln 2, $$ so $$f'(5) = 2^{5} \ln 2 = 32 \ln 2.$$ $\square$

Abbreviated Rules of Differentiation. $$\begin{array}{|l|l|ll|} \hline \text{SR: }& (f \pm g)'=f' \pm g' & \frac{d}{dx}(u \pm v)=\frac{du}{dx} \pm \frac{dv}{dx}\\ \text{CMR: }& (cf)'=cf'& \frac{d}{dx}(cu)=c\frac{du}{dx}\\ \text{PR: }& (fg)'=f'g+fg' & \frac{d}{dx}(uv)=\frac{du}{dx}v+u\frac{dv}{dx}\\ \text{QR: }& \left( \frac{f}{g} \right)'=\frac{f'g-fg'}{g^2} & \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{\frac{du}{dx}v-u\frac{dv}{dx}}{v^2}\\ \text{CR: }& (f\circ g)'=f'g' & \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\\ \text{IR: }& \left( f^{-1} \right)'=\frac{1}{f'} & \frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}\\ \hline \end{array}$$

Example. Find: $$\lim_{x \to 0} \frac{\sin 4x}{\sin 6x} .$$ We have an indeterminate expression $\frac{0}{0}$ and the main has been to resolve it by means of algebra. This time we utilize a limit that has already undergone this transformation: $$\lim_{t \to 0} \dfrac{\sin t}{t} = 1. $$ Algebra is needed to find in our limit $t$ for each $\sin t$: $$\begin{aligned} & = \lim_{x \to 0} \underbrace{\frac{\sin 4x}{4x}}_{t = 4x} \cdot \underbrace{\frac{6x}{\sin 6x}}_{t = 6x} \cdot \underbrace{\frac{4}{6}}_{\text{Make up the difference}} \\ & \text{These limits exist, so apply PR for limits.} \\ & = \lim_{x \to 0} \frac{\sin 4x}{4x} \cdot \lim_{x \to 0} \frac{6x}{\sin 6x} \cdot\frac{4}{6} \\ & = \underbrace{1}_{\text{Since } x \to 0 \text{ means } 4x \to 0} \cdot \frac{1}{\lim_{x \to 0} \dfrac{\sin 6x}{6x}}\cdot\frac{4}{6} \\ & \overset{\text{QR}}{=} \underbrace{1}_{\text{Since } x \to 0 \text{ means } 6x \to 0} \cdot \frac{4}{6}, \quad \text{use the limit again with } t = 6x \\ & = \frac{4}{6} \end{aligned}$$ $\square$

Example. Compute: $$ \left( \frac{x e^{x}}{\sin x}\right)'. $$

The plan: identify

  • the participants -- functions -- and find their derivatives
  • the relations between them -- algebraic operations -- and write down the corresponding differentiation rules.

Functions:

  • $x \Rightarrow (x)' = 1$,
  • $\sin x \Rightarrow (\sin x)' = \cos x$,
  • $e^{x} \Rightarrow (e^{x})' = e^{x}$.

Operations:

  • Multiplication $\leadsto$ PR: $(fg)' = f'g + g'f$;
  • Division $\leadsto$ QR: $\left(\frac{f}{g}\right)' = \frac{f'g – g'f}{g^{2}}$.

Now, find the derivative of $x\cdot e^{x}$ with PR: above set

  • $f = x$ and
  • $g = e^{x}$.

So $$\begin{aligned} f = x & \Rightarrow f' & = 1 \\ g = e^{x} &\Rightarrow g' = e^{x} \end{aligned}$$ Substitute into PR: $$(x e^{x})' = 1\cdot e^{x} + e^{x}\cdot x = e^{x}( 1 + x).$$

Next, find the derivative of the whole with QR: above set

  • $f = x\cdot e^{x}$ and
  • $g = \sin x$.

We already have their derivatives: $$\begin{aligned} f' & = (x e^{x})' = e^x(1 + x)\\ g' & = (\sin x)' = \cos x \end{aligned}$$ Substitute: $$\left(\frac{x e^x}{\sin x}\right)' = \frac{f^{\prime} g – g' f}{g^{2}} = \frac{e^{x}\cdot(1+x)\cdot\sin x - \cos x\cdot x \cdot e^{x}}{(\sin x)^{2}}.$$ $\square$

Example. What is $$(a^{x})',$$ where $a>0$ is an arbitrary base? We'll use $(e^{x})' = e^{x}$. We know the relationship between them: stretch of the graph. So, $$a^{x} = e^{\text{some power?}} = e^{\ln a^{x}}.$$ Here $e^{\ln}$ cancels as they are inverses of each other. How to convert to $e^{..}$? Use the Base Conversion Formula: $$a^{x} = e^{x \ln a}.$$ Now, we are to compute this: $$(a^{x})' = (e^{x\ln a})'.$$ Decomposition first: $$ e^{x\ln a}: \underbrace{x \mapsto u = \ln a}_{\frac{du}{dx}= \ln a} \leadsto \underbrace{y = e^{u}}_{\frac{dy}{dx} = e^{u}}.$$ Apply CR: $$(a^{x})' = \ln a\cdot e^{u} = \ln a\cdot e^{x\ln a} = \ln a\cdot a^{x}.$$ So $$(a^{x})' = a^{x}\ln a \leadsto (e^{x})' = e^{x}.$$ $\square$

Example. We can derive the derivative of cosine from that for sine. What is the relationship between $\sin$ and $\cos$?

Sin vs cos.png

This is what we know from trigonometry: $$\cos x = \sin y, $$ where $x,$ are the two angles of the right triangle. In other words, we have $$y = 90^{\circ} – x \text{ or } y = \frac{\pi}{2} – x.$$ Then, $$\underbrace{\cos x = \sin \left( \frac{\pi}{2} – x \right)}_{\text{flip about } y\text{– axis then shift horizontal}}$$ Use CR to find $\dfrac{d}{dx}\cos x$. First, decomposition: $$\begin{array}{lll} y = \cos x, & u = \frac{\pi}{2} – x, & y = \sin u , \\ & \frac{du}{dx} = -1 &\frac{dy}{du} = \cos u, \end{array} $$ So by CR, we have $$\frac{d}{dx} \cos x = -1 \cdot \cos u = -\cos\left(\frac{\pi}{2} – x\right) = \sin x.$$ $\square$

Example Find: $$\left( \sqrt{\sin x} \right)'.$$ The function is computed in two steps. Indeed, if $$ y = \sqrt{\sin x}$$ then, to compute, we have to do this: $$ x \mapsto u = \sin x \mapsto y=\sqrt{u}.$$ Then $$\frac{du}{dx} = \cos x, \qquad \frac{dy}{du} = \frac{1}{2\sqrt{u}}.$$ Apply CR, then substitute $u = \sin x$, $$\begin{aligned} \frac{dy}{dx} &= \cos x\cdot \frac{1}{2\sqrt{u}} & = \cos x \cdot \frac{1}{2\sqrt{\sin x}}. \end{aligned}$$ $\square$

Example.Find the derivative of: $$z = e^{\sqrt{\sin x}}$$ Three functions this time: We just do them one at a time: $$x \mapsto y = \sqrt{\sin x} \mapsto z = e^{y}. $$ Then $$\begin{aligned} \frac{dy}{dx} & = \cos x\cdot\frac{1}{2\sqrt{\sin x}}\cdot\frac{dz}{dy} = e^{y} \\ \frac{dz}{dx} & = \cos x \cdot \frac{1}{2\sqrt{\sin x}}\cdot e^{y} \\ & = \cos x\cdot \frac{1}{2\sqrt{\sin x}}\cdot e^{\sqrt{\sin x}} \end{aligned}$$ $\square$

Example. Find the derivative of $$f(x)=x^{x}$$ But which formula do we apply? Recall,

  • $(x^{2})' = 2x$, by the power formula
  • $(2^{x})' \neq x2^{x-1}$, an unfortunate misuse of the power formula!

Compare and contrast:

  • $2^{x}$ base fixed, exponent varies -- exponent formula;
  • $x^{n}$ exponent fixed, base varies -- power formula.

The exponent formula? Fail: $$(x^{x})' \neq x^{x}\ln x, \quad\text{ base isn't fixed!}$$ The power formula? Fail: $$(x^{x})' \neq x\, x^{x-1},\quad\text{ exponent isn't fixed!}$$

Recall how we found the derivative of the logarithm: $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$ Similarly, by differetiation: $a^{\log_{a} x} = x$, we obtain: $$\frac{d}{dx} (\log_{a} x) = \frac{1}{x\ln a}$$ The trick is called "logarithmic differentiation". It uses the formula: $$e^{\ln u} = u.$$ For example, to find the derivative of $x^{x}$, substitute $u = x^{x}$. Then $$f(x) = \underbrace{e^{\ln x^{x}}}_{\text{fixed base this time!}} = x^{x}$$

$$\begin{aligned} f(x) & = e^{\ln x} = e^{\ln x} \\ x \to v & = \underbrace{x \ln x}_{\frac{dv}{dx}} \overset{\text{PR}}{=} \ln x + x \frac{1}{x} \to \underbrace{y = e^{v}}_{\frac{dy}{dv}} = e^{v} \\ f^{\prime}(x) & = \frac{dv}{dx}\frac{dy}{dv} = \left( \ln x + x\frac{1}{x} \right)e^{v} \\ & = (\ln x + 1)e^{x\ln x} \\ & = (\ln x + 1) x^{x} \end{aligned}$$

$\square$

Example (rising water). Suppose we have a cylindrical water tank:

  • radius $5$;
  • water is pumped in at $3 \frac{\text{m}^{3}}{\text{min}}$;
  • how fast is the level increasing?
Cylindrical water tank.png

I. Name the quantities:

  • $r$ radius,
  • $V$ volume,
  • $A$ area of the bottom,
  • $H$ height of water,
  • $t$ time.

II. Collect information by rephrasing the data in the problem in terms of these variables:

  • $r=5$;
  • $\frac{dV}{dt}=3$;
  • $\frac{dH}{dt} = ?$

III. Connect these variables (using the geometry of the situation): $$\left. \begin{aligned} A & = \pi r^{2} \\ V &= AH \end{aligned} \right\} \Longrightarrow V = \pi r^{2} H = 25\pi H .$$

IV. Differentiate the equation in (III): $$\begin{aligned} \frac{d}{dt}\left(V\right) &= \frac{d}{dt}\left( 25 \pi H\right), \\ \frac{dV}{dt} & = 25 \pi \frac{dH}{dt}. \end{aligned} $$

V. Substitute the data we know: $$\begin{aligned} 3 &= 25 \pi \frac{dH}{dt} \\ \Longrightarrow\ \frac{dH}{dt} &= \frac{3}{25\pi} \frac{\text{m}}{\text{min}}. \end{aligned}$$ $\square$


Exercise. Find the initial conditions of a free falling object from this data:

Free fall path.png

Exercise. Compute the derivative of: $$x^{x^{2}} = e^{\ln x^{x^{2}}} = e^{x^{2\ln x}}.$$

Example. Find $\frac{dy}{dx}$ from $$ 2xy \sin y = y \cos x .$$ Solution: $$\begin{aligned} \frac{d}{dx}(2xy) & = 2\frac{d}{dx}(xy) \\ & = 2 \left(\frac{d}{dx}(x) \cdot y + \frac{d}{dx}(y) \cdot x \right) \\ & = 2 \left( 1 \cdot y + \frac{dy}{dx} x \right)\quad\Longrightarrow \\ \frac{d}{dx}(\sin y) &= \frac{d}{dx} \left( \underbrace{\sin(y(x))}_{\text{ composition! }} \right) = \cos y\cdot \frac{dy}{dx} \quad\Longrightarrow \\ 2 \left( 1 \cdot y + \frac{dy}{dx} x \right)\sin y+2xy\cos y\cdot \frac{dy}{dx}&=... \end{aligned} $$ $\square$

Example. Differentiate $$ y = \sin(\cos(\tan x )).$$ Decompose and differentiate: $$\begin{array}{llll} &\quad\sin(&&&\quad \cos(&&&\quad \tan x &)&);\\ y &= \sin (v) & \gets &v & = \cos(u) & \gets &u &= \tan(x) &\gets x; \\ \frac{dy}{dv} & = \cos(v) & &\frac{dv}{du} &= -\sin(u) & &\frac{du}{dx} &= \sec^{2}(x) . \\ \end{array}$$ Multiply these, substitute $v = \cos u$ and $u = \tan x$: $$\begin{array}{lll} \frac{dy}{dx} & = \cos(v)&\cdot\left(-\sin(u)\right)&\cdot\sec^{2}(x) \\ &= \cos(\cos(u))&\cdot(-\sin(u))&\cdot\sec^{2}x \\ &= -\cos\cos\tan x &\cdot \sin\tan x &\cdot \sec^{2}x. \end{array}$$ $\square$

Summary of analysis of functions

Stage 1, using $f$:

  • domain,
  • intercepts,
  • symmetry (odd and even functions),
  • periodicity,
  • asymptotes.

Stage 2, using $f'$:

  • monotonicity (increasing/decreasing),
  • extreme points (local max/min),
  • global extreme points.

Stage 3, using $f' '$:

  • concavity,
  • inflection points.

Example. Analyze: $$f(x) = \frac{\sin x}{1 + \cos x}.$$

Stage 1, using $f$.

The domain. Recall that sine and cosine are defined for all $x$. What's left is to make sure there is no division by $0$. Solve for $x$: $$1 + \cos x=0\ \Longrightarrow\ \cos x = -1 \ \Longrightarrow\ x= -\pi + 2k\pi,$$ where $k$ is an integer. Therefore, the domain is: $$\begin{aligned} & = \{x: x \neq -\pi + 2k\pi,\ k \text{ integer } \}\\ & =... \cup (-\pi, \pi) \cup (\pi,3\pi) \cup ... \end{aligned} $$

The $x$-intercepts. Solve $f(x) = 0$, $$\frac{\sin x}{1 + \cos x} = 0\ \Longrightarrow\ \sin x = 0 \ \Longrightarrow\ x = k\pi,$$ where $k$ is an integer.

The $y$-intercept. Set $x=0$, then $$ f(0) = \frac{\sin 0}{a + \cos 0} = 0.$$

Symmetry. Plug in $-x$ and try to get to $f$: $$\begin{aligned} f(-x) &= \frac{\sin (-x)}{1 + \cos(-x)} \\ & = \frac{-\sin x}{1+ \cos x} \\ & = -\frac{\sin x}{1 + \cos x} \\ & = -f(x). \end{aligned}$$ So $f$ is odd.

Periodicity. Sine and cosine are periodic with period $2\pi$: $$\begin{aligned} \sin (x + 2 \pi) &= \sin x, \\ \cos (x + 2 \pi) &= \cos x. \end{aligned}$$ Substitute: $$\begin{aligned} f(x + 2\pi) &= \frac{\sin(x + 2\pi)}{1 + \cos (x + 2pi)} \\ &= \frac{\sin x}{1 + \cos x} \\ &= f(x). \end{aligned}$$ So, $f$ is $2\pi$-periodic.

Asymptotes. Consider now the possibility of vertical asymptotes at those missing point of the domain: $$\begin{aligned} \lim_{x \to -\pi} f(x) &= \lim_{x \to -\pi} \frac{\sin x}{1 + \cos x} \quad {\to \over \to} \frac{0}{0} \\ & \overset{\text{LR}}{=} \lim_{x \to -\pi} \frac{\cos x}{-\sin x} \\ &= \lim_{x\to -\pi} -\cot x = \pm \infty. \end{aligned}$$ So, $x = -\pi + 2k\pi$, for each $k$ integer, are vertical asymptotes.

What about horizontal asymptotes? Evaluate the limit: $$\lim_{x \to \infty} f(x) = \lim_{x\to\infty}\frac{\sin x}{1+\cos x} .$$ But the function is periodic, so no horizontal asymptotes.

Stage 2, using $f'$.

Differentiation: $$\begin{aligned} f'(x) &= \left(\frac{\sin x}{1 + \cos x}\right)' \\ & \overset{\text{QR}}{=} \frac{\cos x (1 + \cos x) - (-\sin x)\sin x}{(1 + \cos x)^{2}} \\ &= \frac{\cos x + \cos^{2}x + \sin^{2}x}{(1 + \cos x)^{2}} \\ & \overset{\text{PT}}{=} \frac{\cos x + 1}{(1 + \cos x)^{2}} \\ &= \frac{1}{1 + \cos x}. \end{aligned}$$ Now solve for $x$: $$\frac{1}{1+\cos x} = 0.$$ Impossible! Hence, no such $x$. What is the sign of $\frac{1}{1+\cos x}$ then? It's positive, $\frac{1}{1+\cos x} >0$, since $\cos x \geq 1$ for all $x$.

Wrong conclusion: $f$ increasing on domain of $f$: $$..., (-\pi,\pi) \cup (\pi,3\pi) \cup ...$$ Positive derivative means increasing behavior holds only for intervals!

Right conclusion: $f$ increases on each of the intervals: $$(-\pi,\pi), (\pi,3\pi), ...$$ On each but not on the whole...

Stage 3, using $f' '$.

Differentiation: $$\begin{aligned} f' '(x) &= \left( \frac{1}{1+\cos x} \right)'\\ & \overset{\text{QR}}{=} \frac{ -\sin x}{ (1+\cos x)^2}. \end{aligned}$$ Next, set $f' '(x) = 0$, and solve for $x$ $$\sin x = 0,$$ so $x = \pi + 2k\pi$, where $k$ is an integer. $$\begin{aligned} f' '(x) & > 0 \text{ on } (-\pi,0), \text{ concave up,} \\ f' '(x) & < 0 \text{ on } (0,\pi), \text{ concave down.} \end{aligned}$$

The end-result is a hand sketch:

Sketch periodic.png

Note that the graph is meant only to verify the data about the function that we have collected. $\square$

Example. Analyze: $$f(x) = xe^{-x^{2}}.$$

Stage 1, using $f$.

Domain: $(-\infty,\infty)$, because $x$, $x^{2}$, $e^{x}$ are defined for all $x$.

$x$-intercepts: solve $xe^{-x^{2}} = 0$ for $x$. Now, $x=0$ works, if $x \neq 0$, then $e^{-x^{2}} = 0$ but $e^{x} \neq 0$, no none here. So $x=0$ is the only $x$-intercept.

$y$-intercept, same.

Periodicity: no.

Symmetry: $f$ is odd: $$\begin{aligned} f(-x) &= (-x) e^{-(-x)^{2}} \\ &= -x e^{-x^{2}} \\ &= -f(x).\end{aligned}$$

Vertical asymptotes: none, because of $x$, $x^{2}$, $e^{x}$ are defined for all $x$ and there is no division.

Horizontal asymptotes: compute two limits $\lim\limits_{x \to \pm\infty} f(x)$. (Note: when $f$ is rational, one limit $\lim_{x \to \infty} f(x)$ is sufficient.) $$\begin{aligned} \lim_{x \to \pm \infty} xe^{-x^{2}} &= \lim_{x \to \pm\infty} \frac{x}{e^{x^{2}}} \\ &\overset{\text{LR}}{=} \lim_{x \to \pm\infty} \frac{(x)^{\prime}}{(e^{x^{2}})'} \\ &= \lim_{x \to \pm\infty} \frac{1}{2xe^{x^{2}}} = 0 \end{aligned}$$

Thus, $y=0$ is a horizontal asymptote at $+\infty$ and $-\infty$.

Stage 2, using $f'$. $$\begin{aligned} f' &= \left( x e^{-x^{2}}\right)'\\ &= e^{-x^{2}} + x\left( e^{-x^{2}} \right)' \\ & = e^{-x^{2}} - 2x^{2}e^{-x^{2}} \\ & = \underbrace{e^{-x^{2}}}_{>0} \underbrace{( 1 - 2x^{2})}_{\text{only look at this}}. \end{aligned}$$ We need the sign of the derivative. We look at and solve $f'(x) = 0$ first: $$1 - 2x^{2}= 0 \ \Longrightarrow\ 2x^{2} = 1 \ \Longrightarrow\ x^{2} = \frac{1}{2} \Longrightarrow\ x = \pm \frac{1}{\sqrt{2}}.$$ These are the only critical points.

Stage 3, using $f' '$. $$\begin{aligned} f' '(x) &= \left( e^{-x^{2}} ( 1 - 2 x^{2} ) \right)' \\ &= -2xe^{-x^{2}} ( 1 - 2x^{2}) + e^{-x^{2}} (-4x) \\ &= e^{-x^{2}} \underbrace{( -2x + 4x^{3} - 4x)}_{\text{cubic polynomial}} \\ & = \underbrace{e^{-x^{2}}}_{>0} \underbrace{2x(2x^{2} - 3)}_{\text{look at this only}} \end{aligned}$$ Solving $f' '(x)=0$, we get $$ x=0, \qquad x = \pm\sqrt{\frac{3}{2}}. $$

We summarize the data in this hand sketch:

Odd function.png

$\square$

Example. Evaluate: $$\lim_{x\to \infty}(\sqrt{x^{2} + x} - x)=?$$ Convert to a fraction by multiplying/dividing by the conjugate: $$\begin{aligned} =\lim_{x\to \infty} \frac{(\sqrt{x^{2} + x} - x)(\sqrt{x^{2} + x} + x)}{(\sqrt{x^{2} + x} + x)} &= \lim_{x\to\infty} \frac{x^{2} + x - x^{2}}{\sqrt{x^{2} + x} + x} \\ &= \lim_{x\to \infty}\frac{x}{\sqrt{x^{2} + x} + x} \qquad \to \frac{\infty}{\infty} \\ &\overset{LR}{=} \lim_{x\to\infty} \frac{1}{\frac{1}{2}\left(x^{2} + x\right)^{-\frac{1}{2}}\cdot\left(2x+1\right) + 1} &???\\ & = \lim_{x\to\infty}\frac{1}{\sqrt{1+\frac{1}{x}}+1} = \frac{1}{2}. \end{aligned}$$ L'Hopital's Rule applied above failed to simplify the denominator. Instead we divided by $x$ as before. $\square$

Example. Approximation via linearization: $$\begin{array}{lll} \sqrt{26}&=\sqrt{26}-5+5\\ &=\left( \sqrt{26}-\sqrt{25} \right) +5\\ &\approx\left(\sqrt{x}\right)'\Big|_{x=25}\cdot(26-25)+5\\ &=\frac{1}{2\sqrt{x}}\Big|_{x=25}\cdot 1+5\\ &=\frac{1}{10}+5\\ &=5.1. \end{array}$$ $\square$

2 Sequences

  • Evaluate the limit if it exists:

$$\lim_{n\to\infty}\frac{(-1)^n}{n}.$$

  • Evaluate the limit: $\lim_{n\to \infty}\ln \left({\frac{n}{n+1}} \right)$.
  • Give an example of a sequence for each of the following: (a) $a_n\to 0$ as $n\to \infty$, (b) $a_n\to 1$ as $n\to \infty$, (b) $a_n\to +\infty$ as $n\to \infty$, (c) $a_n$ diverges but not to infinity.
  • Write a formula for the $n$th term of the sequence:

$$-\frac{1}{2},\ \frac{3}{4},\ -\frac{7}{8},\ \frac{15}{16},\ -\frac{31}{32},\ ...$$


3 Rates of change

  • (a) Suppose during the first $2$ seconds of its flight an object progressed from point $(0,0)$ to $(1,0)$ to $(2,0)$. What was its average velocity and average acceleration? (b) What if the last point is $(1,1)$ instead?
  • Suppose $t$ is time and $x$ is the price of bread. What can you say about its dynamics? Be as specific as possible.
Price of bread over time.png
  • Find the average rate of change for the function given by the following data:

$$\begin{array}{l|c} x&y=f(x)\\ \hline -1&0\\ 0&2\\ 1&3\\ 2&-1\\ 3&-2\\ 4&0\\ \end{array}$$

  • Find the difference quotients for the function given by the following data:

$$\begin{array}{l|c} x&y=f(x)\\ \hline -1&2\\ 0&2\\ 1&3\\ 2&3\\ 3&-2\\ 4&5\\ \end{array}$$

  • Plot the graph of the average velocity for the following position function:
Time vs position via secants.png
  • What are the secant lines of the absolute value function $f(x)=|x|$?
  • (a) State the definition of the average rate of change of a function at point $a$. (b) Find it for the function $f(x)=x^2+3$ at $a=1$ and $h=.5$.
  • The position and the velocity are plotted below. Plot the acceleration.
Time vs position and forms.png
  • The pictured graph represents the number of mosquitoes in a certain area over the period of $150$ days. What happened to (a) the mosquito population and (b) its rate of growth? Explain.
Mosquito population.png
  • Each of these straight lines are drown through two point of the graph. What do they tell us about the function?
Tangent lines.png
  • (a) State the definition of the average rate of change of function $f$ at point $x=a$. (b) Sketch an illustration of this definition for $f(x)=x^2$.
  • The pictured graph represents the altitude (in thousands of feet) of a plane above the ground at time $x$ (in seconds). (a) What does the slope of the curve represent? (b) Describe what happened to the plane.
Decrease and horizontal asymptote.png
  • From the definition, compute the average rate of change for the function $f(x)=x^{2}+1$ at $a=2$ with $h=.2$ and $h=.1$. Explain the difference.
  • (a) Compute the average rate of change for the function $f(x)=3x^{2}-x$ at $a=1$ and $h=.5$. (b) Find the equation of the secant to the graph of $y=f(x)$ corresponding to this average rate of change.
  • The graph of a function $f(x)$ is given below. Estimate the values of the difference quotient $\frac{\Delta f}{\Delta x}$ for $x=0,4,$ and $6$ and $\Delta x=.5$.
Graph for derivative 2.png
  • The graph of a function $f(x)$ is given below. Estimate the values of the difference quotient for $x=2,4,$ and $9 $and $\Delta x=1$.
Estimate derivative from graph.png
  • The graph of a function $f$ is given below. Estimate the values of the difference quotient $\frac{\Delta f}{\Delta x}$ for $x=1$ and $\Delta x=2,\ 1,\ .5$.
Graph for derivative.png
  • The graph of a function $f$ is given below. Estimate the values of the difference quotient $\frac{\Delta f}{\Delta x}$ for $x=0$ and $x=4$ and $\Delta x=1$. Show your computations.
Graph for derivative.png
  • The graph of a function $f$ is given below. Find an equation of the line secant to the graph at $(0,-1)$.
Graph for derivative.png
  • Explain this picture:
Secant lines converge.png
  • The secant line of the sign function are shown below. What do they tell you about the differentiability of the function at $x=0$?
Secant lines of sign.png
  • You have received the following email from your boss: "Tim, Look at the numbers in this spreadsheet. This stock seems to be inching up... Does it? If does, how fast? Thanks. -- Tom". Describe your actions.

4 Limits

  • Explain why the limit $\lim_{x\to 0} \sin\frac{1}{x}$ does not exist.
  • Sketch the graphs of three functions with the three different types of discontinuities. Describe these discontinuities with limits.
  • (a) State the $\varepsilon$-$\delta$ definition of limit. (b) Use the definition to prove that $\lim_{x\to 0}x^2 =0$.
  • (a) State the definition of limit. (b) Use the definition to prove that $\lim_{x\to 0}x^3 \ne 3$.
  • By computing necessary limits, find the vertical asymptotes of the function:

$$f(x)=\frac{x}{(x-1)(x+2)^2}.$$

  • Give an example of a function with two vertical asymptotes: $x=0$ and $x=2$.
  • Give an example of a function with a horizontal asymptote: $y=-1$, and a vertical asymptote: $x=2$.
  • Identify all important features of this graph:
Rational function with several asymptotes.png
  • Express the asymptotes of this function as limits and identify other of its important features:
Rational function with oblique asymptote.png
  • Express the asymptotes of this function as limits and identify other of its important features:
Rational function with oblique asymptote 2.png
  • (a) State the Sandwich Theorem. (b) Give an example of its application.
  • (a) State the Intermediate Value Theorem. (b) Give an example of its application.
  • What is this function?
Sin1x.png
  • True or false: “if $f$ is continuous on $(a,b)$, then $f$ is bounded on $(a,b)$”?
  • True or false: “if $f$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$”?
  • True or false: “if $f$ is continuous on $[a,b)$, then $f$ is bounded on $[a,b)$”?
  • True or false: “if $f$ is continuous on $[a,\infty)$, then $f$ is bounded on $[a,\infty)$”?
  • True or false: “every function is bounded on a closed bounded interval”?
  • True or false: “if a function is continuous on a closed/open interval, then its domain is a closed/open interval”? You have four options to consider.
  • Find the horizontal asymptote of the function:

$$f(x)=\frac{x}{(x-1)(x+2)^2}.$$

  • What can you say about $\lim_{x\to 0}\sqrt{x}$?
  • (a) Give an example of a function with two different horizontal asymptotes. (b) Why can't a rational function have more than one horizontal asymptote?
  • Compute the one-sided limits of the function below at $x=-1$ and $x=3$:

\[ f(x)=\left\{ \begin{align} {}% -x+1 &\quad \text{ if } x<-1;\\ x^{2}+1 &\quad \text{ if } -1\leq x<3;\\ e^x &\quad \text{ if }x>3. \end{align} \ \right. \]

  • By computing a certain limit, find the horizontal asymptote of the function:

$$f(x)=\frac{3x^3-1}{x(5x^2-7)}.$$

  • The base salary of a salesman working on commission is $\$20,000$. For each $\$10,000$ of sales beyond $\$50,000$, he is paid a $\$1,000$ commission. Let $f(x)$ represent his salary as a function of the level of his sales $x$. (a) Sketch the graph of the function. (b) Discuss the continuity of $f$. (b) Discuss the differentiability of $f$.
  • For the function below, find its large scale behavior, i.e., $f(x)\to ...$ as $x\to \pm \infty$:

$$f(x)=\frac{2x^2}{7x^2-x+1}.$$

  • The graph of $f$ is given below. It has asymptotes. Describe them as limits. Hint: use both $+ \infty$ and $- \infty$.
Rational function with 3 asymptotes.png
  • Evaluate the limit below. What is that you've found?

$$\lim_{x\to \infty}\frac{3x^2-7}{5x^2+x}$$

  • (a) Given two functions $f$ and $g$ continuous at $x=a$, does $f+g$ have to be continuous at $x=a$? (a) Given two functions $f$ and $g$ one continuous and the other discontinuous at $x=a$, does $f+g$ have to be continuous or discontinuous at $x=a$? (a) Given two functions $f$ and $g$ discontinuous at $x=a$, does $f+g$ have to be discontinuous at $x=a$?
  • (a) Finish the sentence: “Function $f$ is continuous at $x=a$ if...” (the definition). (b) Use the definition in part (a) to prove or disprove that the function $f$ defined below is continuous at $a=0$ and $a=3$:

\[ f(x)=\left\{ \begin{align} {}% x+1 & \text{ if } x<0;\\ x^{2}+1 & \text{ if } 0\leq x<3;\\ 10 & \text{ if }x>3. \end{align} \ \right. \]

  • Using the $\varepsilon$-$\delta$ definition of limit, prove that

$$\lim\limits_{x\to 1}(-x+1)=0.$$

  • A house has $4$ floors and each floor has $7$ windows. What was the year when the doorman's grandmother died?
  • Show that the theorem about boundedness of a continuous function fails if one of the conditions is omitted: (a) the function isn't continuous, (b) the interval isn't closed, (c) the interval isn't bounded.
  • Illustrate with plots (separately) functions with the following behavior: (a) $f(x)\to +\infty$ as $x\to 1$; (b) $f(x)\to -\infty$ as $x\to 2^+$; (c) $f(x)\to 3$ as $x\to -\infty$.

5 Derivatives

  • If two functions are equal, do their derivatives have to be equal too?
  • (a) State the definition of the derivative of a function at point $a$. (b) Use part (a) to compute the derivative of $f(x)=x^2+3$ at $a=1$.
  • The pictured graph represents the number of mosquitoes in a certain area over the period of $150$ days. What happened to (a) the mosquito population and (b) its rate of growth? Explain.
Mosquito population.png
  • What do these straight lines tell us about the function?
Tangent lines.png
  • (a) State the definition of the derivative of function $f$ at point $x=a$ as a limit. (b) Sketch an illustration of this definition for $f(x)=x^2$ and $a=1$.
  • The pictured graph represents the altitude (in thousands of feet) of a plane above the ground at time $x$ (in seconds). (a) What does the slope of the curve represent? (b) Describe what happened to the plane.
Decrease and horizontal asymptote.png
  • (a) State the definition of the derivative of a function at point $a$. (b) Provide a graphical interpretation of the definition.
  • (a) State the definition of the derivative of a function at point $a$. (b) Use part (a) to compute the derivative of $f(x)=2x^{2}+x-1$ at $a=1$.
  • From the definition, compute the derivative of $f(x)=x^{2}+1$ at $a=2$.
  • From the definition, compute the derivative function of $f(x)=\frac{x}{x+1}$.
  • (a) Compute the derivative of $f(x)=3x^{2}-x$ at $a=1$ from the definition (i.e., as a limit). (b) Find the equation of the line tangent to the graph of $y=f(x)$ at the point corresponding to $a=1$.
  • The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=0,4,$ and $6.$
Graph for derivative 2.png
  • The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=2,4,$ and $9$.
Estimate derivative from graph.png
  • From the definition, compute the derivative of $f(x)=2x^2-1$ at $a=1$.
  • Suppose you have a function $f(x)=\frac{1}{x+1}$. (a) Evaluate this limit: $\lim_{h\to 0}\frac{f(h)-1}{h}.$ (b) What is it that you've found? Illustrate with a picture.
  • The graph of a function $f$ is given below. Estimate the values of the derivative $f'(x)$ for $x=-3,\ 1$, and $5$.
Graph for derivative.png
  • The graph of a function $f$ is given below. Estimate the values of the derivative $f'$ for $x=0$ and $x=4$. Show your computations.
Graph for derivative.png
  • The graph of a function $f$ is given below. Find the equation of the line tangent to the graph at $(0,-1)$.
Graph for derivative.png
  • The graph of function $f$ is given below. (a) At what points is $f$ continuous? (b) At what points is $f$ differentiable?
Continuity -- differentiability.png
  • The graph of a function $f(x)$ is given below. Estimate the values of the derivative $f'(x)$ for $x=1,3$, and $6$.
Graph on grid.png
  • The graph $y=f(x)$ of function $f$ is sketched below (it's not a parabola). Based on the graph, estimate the value of the derivative $f'$ of $f$ for $x=0$ and $x=5$. What can you say about $f' '$?
Parabola on grid.png

6 Features of graphs

  • The graph of $f$ is given below. Find all the asymptotes of the function and describe them as limits.
Rational function graph.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f′$ in the space under the graph of $f$. Identify all important points and features on the graph.
Rational function graph 2.png
  • Sketch the graphs of functions with these features: (a) $f$ has a local maximum at $x=2$; (b) $g$ has a vertical asymptote $x=1$; (c) $h$ has a horizontal asymptote $y=-1$.
  • Sketch the graph of function $y=f(x)$ satisfying the following conditions: $\lim_{x\to 2^-}f=1$, $\lim_{x\to 2^+}f=3$, $f$ is increasing on $(-1,0)$, $\lim_{x\to -\infty}f=-1$, $\lim_{x\to +\infty}f=\infty$.
  • The graph of function $f$ is given below. List at least five of its main features.
Rational function graph.png
  • A sketch of the graph of a function $f$ is given below. Provide the important limits of $f$ that describe its behavior.
Graph with discontinuities and asymptotes.png
  • A sketch of the graph of a function $f$ is given below. List at least five of its main features.
Graph with discontinuities and asymptotes.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative 2.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative 3.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative 4.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative 5.png
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
Sketch the graph of the derivative 6.png
  • (a) Show that one of the functions below is the derivative of the other. (b) Explain how the horizontal asymptote of the derivative affects the graph of the function. (c) What about the horizontal asymptote of the function vs. the graph of the derivative?
Function and its derivative.png
  • Describe the behavior of the function plotted below:
Describe the behavior.png
  • Describe the concavity of this function:
Graph on grid.png
  • The graph of $f$ is given below. Completely describe the behavior of the function by using such words as “increasing/decreasing”, “concave up/down”, “max/min”, “asymptotes”, etc.
Rational function with several asymptotes 2.png
  • The graph of $f$ is given below. For what values of $x$ are $f(x),\ f'(x),\ f' '(x)$ positive, negative or zero? Fill in the blanks with $+$, $-$, or $0$:

$$\begin{array}{r|ccc} x&f(x)&f'(x)&f' '(x)\\ \hline -1.5\\ 0\\ 1\\ 2\\ 2.5 \end{array}$$

Signs of derivatives.png
  • The graph of $f$ is given below. For what values of $x$ are $f(x),\ f'(x),\ f' '(x)$ positive, negative or zero? Fill in the blanks with $+$, $-$, or $0$:

$$\begin{array}{r|ccc} x&f(x)&f'(x)&f' '(x)\\ \hline -1.5\\ 0\\ 1\\ 2\\ 2.5 \end{array}$$

Signs of derivatives 2.png
  • Sketch the graph of $f$ with these values of $f(x),\ f'(x),\ f' '(x)$:

$$\begin{array}{r|ccc} x&f(x)&f'(x)&f' '(x)\\ \hline -1&+&+&+\\ 0&-&0&-\\ 1&+&&-\\ 2&0&-&\\ 3&-&+& \end{array}$$

  • Sketch the graph of a function $f$ that is continuous on $[1,5]$ and has global minimum at $1$, global maximum at $5$, local maximum at $2$, and local minimum at $4$.
  • Sketch the graph of a function $f$ with the following features: (a) it has a removable discontinuity at $x=-1$; (b) it has a vertical asymptote $x=1$; (c) it is continuous but not differentiable $x=3$; (d) it is differentiable everywhere else; (e) it has no horizontal asymptotes.
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$.
Graph with 2 maxima.png
  • The graph of the derivative $f'$ of function $f$ is given below. Sketch a possible graph of the function $f$ itself in the space under the graph of $f'$. Identify all important points on the graph.
Function from derivative.png
  • The graph of the derivative $f'$ of function $f$ is given below. Sketch a possible graph of the function $f$ itself in the space under the graph of $f'$. Identify all important points on the graph.
Function from derivative 2.png
  • The graph of the derivative $f'$ of function $f$ is given below. Sketch a possible graph of the function $f$ itself in the space under the graph of $f'$ under the assumption that $f(0)=0$.
Function from derivative 3.png
  • Suppose you are towing a trailer-home. During the first few minutes, every time you look at the rear view mirror you can see only the lower part of the home. Later, every time you look you can see only the top part. Discuss the profile of the road.
  • The graph of function $f$ is given below. Sketch the graph of the derivative $f'$ of $f$:
Graph and local extrema.png
  • Sketch the graph of a differentiable function $f$ that is continuous on $(-\infty ,6]$ and has an global maximum at $-1$, a local maximum at $2$, a local minimum at $5$, an inflection point at $4$, and a horizontal asymptote $y=1$.
  • (a) Analyze the first and second derivatives of the function $f(x)=x^4-2x^2$. (b) Use part (a) to sketch its graph of $f$.
  • (a) Finish the statement “If $h'(x)=0$ for all $x \in (a,b)$ then...”. (b) Finish the statement “If $f'(x)=g'(x)$ for all $x$ in $(a,b)$ then...”. (c) Use part (a) to prove part (b).
  • Find the vertical and horizontal asymptotes of the function and describe

them as limits: \[ f(x)=\dfrac{2x^{2}}{x^{2}-1}. \]

7 Linearization

  • Use linear approximation of $f(x)=\sin x$ to estimate $\sin .02.$
  • Find the linear approximation of $f(x)=\ln x$ at $a=1$. Use it to estimate $\ln .99$.
  • Find the linear approximation of $f(x)=\sqrt{x}$ at $a=1.$ Use it to estimate $\sqrt{1.1}.$
  • Find the linear approximation of $f(x)=\sin 3x$ at $a=0.$ Use it to estimate $\sin -.02.$
  • Find the linear approximation of $f(x)=\sqrt{1+3x}$ at $a=0.$ Use it to estimate $\sqrt{1.03}.$
  • Find the linear approximation to estimate $\sqrt[3]{26.9}.$
  • Find the linear approximation of $f(x)=x^{1/3}$ at $a=1$. Use it to estimate $1.1^{1/3}$.
  • Use linear approximation to estimate $\sin \pi/2$.
  • Use linear approximation to estimate $\sin \pi/4$.

8 Models

  • A cup of hot chocolate has temperature $80$ degrees in a room kept at $20$ degrees. After an hour the chocolate cools to $60$ degrees. (1) Assuming Newton's Law of Cooling, what is the temperature of the chocolate after another hour. (2) Provide the formula for Newton's Law of Cooling and explain.
  • The velocity of the object at time $t$ is given by $v(t)=1+3t^{2}.$ If at time $t=1$ the object is at position $x=4,$ where is it at time $t=0$?
  • The graphs of the velocity and the position of a moving object are shown below. Describe what is happening.
Velocity and position.png
  • The graphs of the velocity and the position of a moving object are shown below. Describe what is happening.
Velocity and position 2.png
  • Suppose the altitude, in meters, of an object is given by the function $t^2+t$, where $t$ is time, in seconds. What is the velocity when the altitude is $12$ meters?
  • The velocity of the object at time $t$ is given by $v(t)=1+e^{t}.$ If at time $t=0$ the object is at $x=2,$ where is it at time $t=1$?
  • The acceleration of an object at time $t$ is given by $a(t)=3t.$ If at time $t=1$ the velocity of object is at $v(1)=-1,$ what is it at time $t=0$?
  • Suppose $s(t)$ represents the position of a particle at time $t$ and $v(t)$ its velocity. If $v(t)=\sin t-\cos t$ and the initial position is $s(0)=0,$ find the position $s(1).$
  • Suppose the speed of a car was growing continuously following the rule $55+5t$ per hour, where $t$ is the number of hours passed since it was $250$ miles away from a city. How far is it from the city after $3$ hours of driving towards it?
  • Let $x$ represent the time passed since the car left the city. The table below tells for what values of $x$ the velocity and the acceleration of the car are positive, negative, or zero. Let $f(x)$ represent the distance of the car from the city. Sketch the graph of $f$.

$$\begin{array}{c|cc} x&\text{ velocity }&\text{ acceleration }\\ \hline 0&0&+\\ 1&+&-\\ 2&0&-\\ 3&-&- \end{array}$$

  • The population of beetles in a certain location is projected to grow at the rate $10,000+2,000x^{2}$ per month, where $x$ is the number of months passed since the beginning of this year. What is the projected population at the end of December next year if the current population is $1,000,000$?
  • The graph of the velocity $f$ as a function of time is given below. Sketch the graph of the location function $F$ that satisfies $F(0)=1$.
Graph with 2 maxima.png
  • The height of the ball (in feet) $t$ seconds after it is thrown is given by $f(t)=-16t^2+8t+6$. Explain the meaning of the numbers $-16,8,6$.
  • The graph of the velocity of a car is given below. Plot the graph of the function representing the distance of the car from the starting point.
Velocity of a car.png
  • Suppose the speed of a car was changing continuously following the rule $60-t^{2}$ per hour, where $t$ is the number of hours passed since noon. Find the average speed of the car between $1$ pm and $3$ pm.
  • Suppose the altitude, in meters, of an object is given by the function $$y=t^2+t,\ t\ge 0,$$ where $t$ is time, in sec. What is the velocity when the altitude is $12$ meters?
  • The population of a city declines by $10\%$ every year. How long will it take to drop to $50\%$ of the current population?
  • The area of a circle is increasing at a rate of $5$ square centimeters per second. At what rate is the radius of the circle increasing when the area is $2$ cm?
  • Find the initial conditions of a free falling object from this data:
Free fall path.png

9 Information from the derivatives

  • (a) Analyze the function $f$ given below and its derivatives. (b) Use part (a) to sketch the graph $f$:

$$f(x)=\frac{x^2+7x+3}{x^2}.$$

  • Find all critical points of the function $f(x)=2x^3-6x+7$.
  • Suppose the derivative of a function $f$ is $f'(x)= \ln x +\ln x^2$. (a) On what intervals, if any, is f increasing? (b) On which intervals, if any, is $f$ concave down? Hint: simplify first.
  • Find all local maxima and minima of the function $f(x)=x^{3}-3x-1$.
  • (a) Analyze the first and second derivatives of the function $f(x)=x^{4}-2x^{2}$. (b) Use part (a) to sketch its graph of $f$.
  • Suppose the functions that follow are differentiable. (a) Finish the statement “If $h'(x)=0$ for all $x$ in $(a,b)$ then...”. (b) Finish the statement “If $f'(x)=g'(x)$ for all $x$ in $(a,b)$ then...”. (c) Use part (a) to prove part (b).
  • (1) State the Mean Value Theorem and illustrate it with a sketch. (b) Use the theorem to prove that if two functions have equal derivatives, then they differ by a constant.
  • Use Newton's method for $f(x)=x^{5}+2$ with $x_{1}=-1$ to find $x_{3}$. What is the meaning of what you've found?
  • Sketch the graph of the function $f(x)=\sqrt{x}e^{-x}$. Justify the graph by studying the derivatives of $f$.
  • (1) State Rolle's Theorem and illustrate it with a sketch. (b) Quote and state the theorem(s) necessary to prove it. What theorem follows from it?
  • Compute the first and second derivatives of the function $f(x)=x^{3}-3x$ and use them to sketch its graph.
  • (a) Finish the statement “If $h'(x)=0$ for all $x$ in $(a,b)$ then...”. (b) Use the theorem in part (a) to prove that if two functions $f,g$ have equal derivatives, then they differ by a constant.
  • Sketch the graph of the function given below. Provide justification for each feature of the graph:

$$f(x)=\frac{x^2+7x+3}{x}.$$

  • (a) State the Mean Value Theorem. (b) Verify that the function $f(x)=\frac{x}{x+2}$ satisfies the hypotheses of the theorem on the interval $[1,4]$.
  • Sketch the graph of the function $f(x)=x^{4}-x^2$. Provide justification for each feature of the graph.
  • Find global maxima and minima of the function, $f(x)=x^{3}-3x$ on the interval $[-2,10]$.
  • (a) State the Mean Value Theorem. (b) Give an example of its application.
  • Find the local maximum and minimum points of the function $f(x)=x^3-3x$.
  • If every point on the graph of $y=f(x)$ is a critical point, what does the graph look like?
  • The graph of function $f$ is given below. (a) At what points is $f$ continuous? (b) At what points does the derivative of $f$ exist?
Continuity and derivative.png
  • (a) Compute the derivative of $f(x)=3x^{2}-x$ at $a=1$ from the definition (i.e., as a limit). (b) Find the equation of the line tangent to the graph of $y=f(x)$ at the point corresponding to $a=1$.

10 Computing derivatives

  • Calculate the derivative of

$$f(x) = \frac{x^2}{x^2-1}.$$

  • Represent this function $h(x)= \sqrt{x^2-1}$ as the composition of two functions. Find its derivative.
  • Calculate the derivative of $f(x) = x^{\pi} + \pi^{x} + x + \pi$ indicating the rules you use.
  • Calculate the derivative of $f(x) = \log_{\pi}x + \log_x \pi$ indicating the rules you use.
  • Suppose $f(1)=3$ and $f'(1)=2$. Use this information to ill in the blanks:

$$\left( f^{-1}(\quad )\right)'=\quad.$$

  • Differentiate this:

$$g(t)=t \cos t \sin t.$$

  • Differentiate:

$$\frac{\ln(\sin x)}{x}.$$

  • Compute the derivative of $f(x)=e^{x^2+3x}$.
  • Evaluate $\frac{d}{dx}\left( \sin x \cdot e^{x+1} \right)$.
  • Evaluate $\frac{d}{dx}\left( \cos t +e^t \right)$. Hint: watch the variables.
  • Evaluate $\frac{dy}{dx}$ for $y=\sin e^{2x}.$
  • Evaluate the derivative of of $f(x)=xe^{\sin x}.$
  • Suppose $f'(1)=2$, $g'(2)=3$, and $h'(1)=6$, where $h=g\circ f$. What is $f(1)$?
  • Is it possible that both $F(x)$ and $F(2x)$ are both antiderivatives of some function $f$?
  • Is $\sin x +3x$ an antiderivative of $\cos x^2$?
  • Is it possible that both $F(x)$ and $F(2x)$ are both antiderivatives of some non-zero function $f$?
  • Evaluate the derivative of $f(x)=x^{2}e^{x}$.
  • Find the second derivative of $h(x)=x^{2}+x+1.$ What does it tell you about the shape of the graph of $f?$
  • Find the second derivative of $h(x)=2x^{\pi }$.
  • Compute the derivative of $f(x)=\ln (3x+2)$.
  • Find the second derivative of $h(x)=xe^{x}$.
  • Find the derivatives of the functions: (a) $3x^{e}+e^{\pi}$, (b) $7\ln x+(1/x)-\ln 2.$
  • Differentiate

$$g(t)=\sqrt{x} \cos x.$$

  • Evaluate $\frac{dy}{dx}$ for

$$y=\sqrt{e^{x}}.$$

11 Implicit things

  • Find the slopes of the tangent lines to the ellipse $x^{2}+2y^{2}=1$ at the points where it crosses the diagonal line $y=x$.
  • Use implicit differentiation to find an equation of the line tangent to the curve $x^{1/2}+xy=2$ passing through the point $(1,1)$.
  • Evaluate $\frac{dy}{dx}$ for $y=\sin \cos (-x)$.
  • Find an equation of the line tangent to the curve $xy=1$ passing through the point $(1,1).$
  • Suppose $x\sin y+y^{2}=x$. Find $\frac{dy}{dx}$.
  • A ladder $10$ ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at the rate $1$ ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6$ ft from the wall?
  • The area of a circle is increasing at a rate of $5$ cm$^{2}$/sec. At what rate is the radius of the circle increasing when the area is $2$ cm?
  • Use implicit differentiation to find an equation of the line tangent to the curve $x^{2}+y^{2}=x$ passing through the point $(0,0)$.
  • Two cars start from the same point. One travels north at $60$ mi/h and the other travels east at $25$ mi/h. How fast is the distance between them increasing two hours later?
  • The volume of a cube is increasing at a rate of $24$ cm$^{3}$/min. How fast is the edge of the cube increasing when the volume is $8$ cm$^{3}?$
  • Suppose $xy+x^2y^3=1$. Find $\frac{dy}{dx}$.
  • Find an equation of the line tangent to the curve $x\sin y=x$ at the point $(1,\pi /2)$.
  • Use implicit differentiation to find an equation of the line tangent to the curve $3x+2y=7$ passing through the point $(1,2)$.

12 Optimization

  • Find the point on the parabola $y^2=2x$ that is closest to the point $(1,4)$.
  • Find the point on the circle $(x-1)^2+(y-2)^2=3$ that is closest to the origin.
  • Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius $1$.
  • Find the point on the parabola $y^2=2x$ that is closest to the point $(2,2)$.
  • A piece of wire $10$ m long is cut into $2$ pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) a minimum?
  • Find the point on the line $y=1-2x$ that is closest to the origin.
  • Find the point on the line $y=-2x$ that is closest to the origin.
  • A farmer has $100$ yards of fencing. What are the dimensions of an enclosure that has the largest area?
  • It is known that a farmer with $100$ yards of fencing should build a $25$-by-$25$ yard enclosure in order to have the largest area. What if he has $200$ yards?
  • A farmer has $100$ yards of fencing. What are the dimensions of an enclosure that has the largest perimeter?
  • Find two numbers $x,y$ whose sum is $2$ and whose product is a maximum.
  • Set up but do not solve the optimization problem for the following situation: “Among all rectangles inscribed in a circle of radius 1, find the one with the largest area”.
  • Set up and solve the optimization problem for the following situation: “Find the point on the line $y=\pi$ that is closest to the origin.”
  • Find the global maximum and minimum points and values of the function $f(x)=2x^{3}-6x+5$ on the interval $[-2,5].$
  • A farmer with $750$ ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?
  • Find two positive integers such that the sum of the first number and four times the second number is $1000$ and the product as large as possible.
  • Find two positive numbers $x,y$ whose product is $100$ and whose sum is a minimum.
  • Set up but do not solve the optimization problem for the following situation: “If $1200$ cm$^{2}$ of material is available to make a box with a square base and an open top, find the largest possible volume of the box.”
  • Set up but do not solve the optimization problem for the following situation: “A poster is to have an area of $180$ in$^{2}$ with $1$-inch margins at the bottom and the sides and a $2$-inch margin at the top. What dimensions will give the largest printed area?”
  • Find a positive number such that the sum of the number and its reciprocal is as small as possible.
  • Set up but do not solve the optimization problem for the following situation: “Among all right triangles with area $10$, find the one with the smallest perimeter”.
  • If an open box is to be made from a tin sheet $8$ in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made.
  • A farmer wants to fence an area of $1.5$ million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

13 Other

  • Indicate which the following statements below is true or false (no proof necessary):
    • 1. If the function $f$ is increasing then so is $f^{-1}.$
    • 2. The exponential function has an asymptote.
    • 3. If $f'(c)=0$ then $c$ is a local maximum or a local minimum of $f$.
    • 4. If a function is differentiable then it is continuous.
    • 5. If two functions are equal, their derivatives are also equal.
    • 6. If two functions are equal, their anti-derivatives are also equal.