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# Calculus exercises: advanced

### From Mathematics Is A Science

- Represent vector $<1,1>$ as a linear combination of vectors $<1,2>$ and $<-1,3>$.

- Is the union of two linear subspaces always a linear subspace? Prove or provide a counterexample.

- For the linear equation $$x_1+2x_2+x_3=0$$ find two vectors $U$ and $V$ such that every solution $X=<x_1,x_2,x_3>$ can be written as a linear combination of $U$ and $V$.

- Solve the following system, expressing the solution in vector form: $$\begin{array}{lll}x_2-x_3=1,\\x_1+x_2+x_3=2.\end{array}$$

- Let $S={\rm span \hspace{3pt}} \{V \}$, where $V=<2,-1>$. (a) Compute the homogeneous linear equation in $x_1,x_2$ that determines $S$. (b) Suppose $A$ is an affine subspace parallel to $S$ and passing through $w=(3,0)$. Compute the linear equation that determines $A$.

- Prove that any four vectors in $R^3$ are linearly dependent. Use the definition of linear independence and properties of $R^3$.

- Prove the homogeneity property of $||x||$.

- Find the projection of $V=<1,2,3>$ onto the plane spanned by $<1,1,1>$ and $<0,1,2>$.

- Find all unit vectors perpendicular to the plane spanned by $<-1,1,0>$ and $<0,1,1>$.

- Prove that the function $f(X)=||X||$ is or is not linear.

- For the linear map $L(x_1,x_2)=<3x_1+x_2,x_1-x_2>$ find the basis of the null space (the kernel).

- Compute the composition of the functions $L(x_1,x_2)=<3x_1+x_2,x_1-x_2>$ and $f(x_1,x_2)=2x_1+x_2$.

- Suppose $L:R^2 \rightarrow R^2$ is linear and suppose $L(e_1)=e_1-3e_2,\ L(e_2)=-2e_1+e_2$. Write $L$ as a $2 \times 2$ matrix.

- Write the equation of the plane which passes through the origin and is perpendicular to the line $x=0,y=t,z=2t$.

- Find either the highest or the lowest point (whichever applicable) of the surface given by the equation $z=f(x,y)=2x^2+y^2-2x-y+1$.

- Measure the volume of the solid lying above the rectangle $R$ in the $xy$-plane consisting of all points $(x,y,0)$ with $0 \leq x \leq 2,0 \leq y \leq 2$ and bounded from above by the surface given by $z=3y^2+4xy$.

- Go with the vector field $$F(x,y)=\frac{1}{\sqrt{x^2+y^2}}<y,-x>, \ (x,y)\neq (0,0). $$ Choose a few points on the plane and pencil in the field vector $F(x,y)$ with tail at $(x,y)$. Pencil in a few typical trajectories of the field.

- Measure the net flow across the curve $y=x^2-1,0 \leq x \leq 1$, of a fluid whose velocity is given by a vector field $F(x,y)=<y,2x>$.

- Here is a plot of a few level curves of a function $F(x,y)$ with a minimizer at $(1,0)$ and a maximizer at $(-1,0)$. Sketch a few trajectories of the vector field $\operatorname{grad}F(x,y)$.

- Let $$F(x,y)=<\sin (xy)+xy\cos (xy),x^2\cos (xy)>. $$ Show that $F$ is a gradient field.

- Suppose $$\operatorname{grad}f(x,y)=(2xy+y^3,x^2+3y^2x).$$ Find $f$.

- Suppose $$f(x,y)=\sin x \cdot \ln (\cos y+1). $$ Find the flow of the vector field $F(x,y)=\operatorname{grad}f(x,y)$ along any curve from $(\frac{\pi}{2},0)$ to $(\frac{\pi}{2},\frac{\pi}{2})$.

- Let $A$ be the area of that part of the surface $z=x^2$ that lies above the unit circle. Write $A$ as a double integral. Do not evaluate.

- Use an appropriate change of variables to find the area of the first-quadrant region bounded by the curves $xy=1,\ xy=2$ and $xy^2=2,\ xy^2=4$.

- Find the volume conversion factor of the transformation $x(u,v,w)=e^{uv},\ y(u,v,w)=uv^2w,\ z(u,v,w)=vw$.

- Use Gauss's formula to evaluate the flow of the vector field $$F(x,y,z)=(xy,x^2+\frac{1}{2}y^2,yz)$$ across the surface of the solid body consisting of all points $(x,y,z)$ with $-1 \leq x \leq 1,|y| \leq 2-x^2,0 \leq z \leq 3$.

- True or false?
- a. Every closed set is bounded.
- b. The empty set is path-connected.
- c. If the partial derivatives exist then the function is differentiable.
- d. Every continuous $1$-dimensional vector field is conservative.
- e. If $C$ is a vertical segment in $R^2$ and $F(x,y)=(\sin(xy),0)$ for all $x,y$, then

$$\displaystyle\int_{C}F(X) \cdot dX=0.$$

- For the following functions $F$ and sets $D$, sketch $F(D)$:
- a. $F(x)=\sin x,\ D=[0,\frac{\pi}{2}) \cup [{\pi},2{\pi}];$
- b. $F(x,y)=x^2+y^2,\ D=\{(x,y):|x| \leq 1,\ |y|\leq 1\};$
- c. $F(x)=(2\cos x,2\sin x),\ D=[0,\frac{\pi}{2}];$
- d. $F(x,y)=(x+y,x-y),\ D=\{(x,y):0\leq x\leq 1,\ 0\leq y \leq 1\}.$

- Find the matrix of the total derivative of $F(x,y)=(x\sin y,x-y)$ at $(1,0).$

- Suppose $f:{\bf R}^2 \rightarrow {\bf R}^2$ is a differentiable vector function such that the matrix of $F'(0,0)$ is the identity matrix. Find the total derivative of the vector function $H(x,y)=F(xy-1,x+y-2)$ at $(1,1)$.

- Find the tangent plane to the parametric surface $f(u,v)=(u-v,v^2,u)$ at the point $(0,0,0)$.

- Evaluate the integral $$\displaystyle\iiint_{B}xy\,dV,$$ where $B$ is bounded by the planes $x=0$, $y=0$, $y=1-x$, $z=0$, $z=1$.

- (a) Present the general formula of change of variables $T$ in an integral. (b) For the case of affine $T$, provide the Riemann sum for the integral and an illustration for the theorem.

- Compute $$\displaystyle\int_{C}xy\,dx+y^2\,dy,$$ where $C$ is the half of the unit circle from $(0,-1$) to $(0,1)$.

- (a) State Stokes' Theorem for “simple regions”. (b) Use part (a) to compute the area of a circle.

- Suppose $$F(x,y)=<y\cos (xy),x\cos (xy)>.$$ Find a potential function of $F$.

- Suppose $C$ is a closed curve that makes one loop around the origin. Find the work of the force $$F(x,y)=\left(\frac{1}{x^2+y^2} \right)<-y,x>$$ along $C$. Justify.

- Suppose $f$ is a differentiable function with $f'(0)=3$. Find the derivative of $g(x,y)=f(x^2+x+y^2+y)$ at $(0,0)$.

- (a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\bf R}^3$.

- For the linear map $L(x_1,x_2) = (3x_1 + x_2, -3x_1 - x_2)$ find the basis of the image space.

- Prove the formula: $X\cdot Y=||X||\,||Y|| \cos \theta$ in $R^2$.

- Using the $\epsilon$-$\delta$ definition, prove that the function $f(x)=||x||,\ x \in R^n$, is continuous.

- Suppose $A$ and $B$ are path-connected subsets of $R^n$ and suppose $a \in A \cap B$. Prove that $A \cup B$ is path-connected.

- Classify these subsets of ${\bf R}^3$ as closed, open, bounded, path-connected or not (no proof required):
- a. $\{(x,y,z):x^2+y^2+z^2=1$ and $0<x<5\},$
- b. the complement of the union of the three axes,
- c. $\{(x,y,z):\ x^2+y^2=1$ and $z^2=1\}$.

- Give an example of a set $S$ and a point $p\in S$ such that $p$ is a limit point of $S$ and but not an interior point. Explain.

- Suppose $f,g:\mathbf{R}\rightarrow \mathbf{R}$ are continuous functions. Let the map $H:\mathbf{R}\rightarrow \mathbf{R}^{2}$ be given by $H(x)=(f(x),g(x)).$ Prove that $H$ is continuous.

- True or false?

- The intersection of two linear subspaces is a linear subspace.
- The empty set is a linear subspace.
- The product of two linear functions is a linear function.
- The acceleration is always perpendicular to the velocity.
- There is only one natural parametrization of a straight line.

- Give an example of:

- a curve with curvature equal to $0$,
- a parametric curve that traces a circle $3$ times clockwise,
- a linear subspace in ${\bf R}^3$ with no vectors perpendicular to it,
- a system of $2$ linear equations with no solutions,
- a system of $2$ linear equations with exactly $2$ solutions.

- (a) Give the definition of a basis of a linear space. (b) Show that the vectors $<1,0,0>,\ <1,1,0>,\ <1,1,1>$ form a basis of ${\bf R}^3$.

- Using the $\varepsilon$-$\delta$ definition, prove that the composition of two continuous functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{m}$ and $g:\mathbf{R}^{m}\rightarrow\mathbf{R}^{p}$ is continuous.

- Suppose $f,g:B\rightarrow \mathbf{R}$ are continuous functions, where $B$ is a disk in $\mathbf{R}^{2},$ such that

\begin{equation*} \begin{array}{l} \text{(1) }f(x)\geq g(x)\text{ for all }x\in B, \\ \text{(2) }\int_{B}f(x)dA=\int_{B}g(x)dA. \end{array} \end{equation*} Using the properties of the integral and continuity, show that $f(x)=g(x)$ for all $x\in B$.

- From the definition, prove that $f(x)=x^3$ is continuous at $x=1$.
- From the definition, prove that any subsequence of a convergent sequence converges.

- Is it possible to have a function $f$, a point $a$, and a sequence $\{x_n\}$ convergent to $a$ such that
- $\lim_{x\to a} f(x)$ does not exist but
- $\lim_{n\to \infty} f(x_n)$ does?

- From the definition, prove that l.u.b. of the set $S=\{x=1-1/n: n\in {\bf N}\}$ is $1$.

- From the definition, prove that if $f$ is continuous at $a$ then so is $-f$.

- Suppose $$\lim_{n\to \infty} x_n=\infty \text{ and } \lim_{n\to \infty} y_n=-\infty.$$ What are the possible outcomes for $$\lim_{n\to \infty} (x_n+y_n)?$$ Give an example for each.

- (a) State the definition of the Riemann sum. (b) Use part (a) to explain the expression below:

$$0^2\cdot .25+.25^2\cdot .25+.5^2\cdot .25+.75^2\cdot .25.$$

- From the definition, prove that $f(x)=x^3$ is uniformly continuous on $[0,1]$.

- Prove: $$ \lim _{x\to 0}x\sin\frac{1}{x}=0.$$

- (a) State theorems that relate differentiability, continuity, and integrability to each other. (b) Provide examples that relate differentiability, continuity, and integrability to each other.

- From the definition, prove that l.u.b. is unique.

- State the Bolzano-Weierstrass Theorem and provide an example that shows that the condition of the theorem cannot be removed.

- Prove that every subsequence of a convergent sequence converges.

- From the definition, prove that l.u.b. of the set $S=\{x=-1/n: n\in {\bf N}\}$ is $0$.

- Suppose $d$ is the standard Euclidean metric of ${\bf R}^2$ while $d_1$ is the taxicab metric: $$d_1\left( (x,y),(u,v) \right)=|u-x|+|v-y|.$$ Do one of the two: (a) prove that if a sequence converges with respect to $d$, it also converges with respect to $d_1$, or (b) prove the converse.

- (a) State the definition of a differentiable function $f:{\bf R}^N\to {\bf R}$. (b) Prove that every differentiable function is continuous.

- (a) State the Mean Value Theorem for dimension $N$. (b) Derive from it the Mean Value Theorem for dimension $1$.

- The formula for the Taylor polynomial of $n$th degree of function $f$ at $a\in {\bf R}^N$ is: $$T_n(x)=\sum_{|\alpha|\le n}\frac{1}{\alpha !}D_\alpha f(a)(x-a)^\alpha.$$ (a) Explain the terms in the formula. (b) Compute $T_2$ for $f(x,y,z)=x^2e^yz$ at $0$.

- Use the Sandwich Theorem to prove that $\lim_{x\to 0} x^n=0$ for every positive integer $n$.

- State the definition of the one-sided limit and state the basic theorems about it.

- From the definition, show that $f(x)=x^{2}-1$ is uniformly continuous on $[0,1]$.

- State the Nested Intervals Theorem and provide examples that show that the conditions of the theorem cannot be removed.

- State and prove the theorem about boundedness of continuous functions.

- Prove that any subsequence of a convergent sequence converges.

- State the Heine-Borel Theorem and provide two examples that show that the conditions of the theorem cannot be removed.

- (a) State Rolle's Theorem and the Mean Value Theorem. (b) Prove one of them.

- Discuss the differentiability of the functions $x\sin(1/x)$ and $x^2\sin(1/x)$.

- From the definition of Darboux integral, prove that $\int_a^b c\,dx=c(b-a)$, where $c$ is a constant.

- State and prove the Fundamental Theorem of Calculus.

- Is it possible that $|f|$ is integrable on $[0,1]$ but $f$ is not?

- (a) State the definition of the limit of a function and three most important theorems about it. (b) Prove one of these theorems.

- Show that if a function is differentiable at a point, then it is continuous at that point.

- Prove that if $f$ is continuous on $[a,b]$ then it is integrable on $[a,b]$.

- Discuss the continuity, differentiability, and integrability of the function the Dirichlet function.

- (a) State the Root Test and the Ratio Test. Give examples of their application. (b) Prove one of them.

- State and prove the theorem about the interval of convergence of power series. Give examples of specific series for each case of the theorem, i.e., for each different type of interval.

- Prove that the uniform limit of a sequence of continuous functions is continuous.

- Show that the sequence $f_{n}(x)=x^{n}$ converges for each $x\in [0,1]$ but the convergence is not uniform. What happens to the sequences of the derivatives and the antiderivatives of $f_{n}?$

- Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.

- Prove that an open ball in a metric space is an open set.

- Prove that a compact set in a metric space is bounded and closed.

- Suppose $S,T$ are metric spaces and $f,g:S\rightarrow T$ are continuous functions. Prove that the set $\{x:f(x)=g(x)\}$ is closed in $S.$ What can you say about $\{x:f(x)\ne g(x)\}$?

- State and prove the fundamental lemma of differentiation for functions of two variables.

- State the definition of a differentiable functions of two variables. Give an example of a function such that both partial derivatives of $f$ exist at $x=a$ but $f$ is not differentiable.

- State the extension of the Mean Value Theorem to functions functions of several variables.

- Give example of such a function of two variables that $f$ is not continuous at $(0,0)$ but both partial derivatives exist at $(0,0).$

- State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.

- Describe Newton's method. Give an example of a function for which the method does not apply.

- Let $S$ be a complete metric space. Then every subset $A$ of $S$ is also a metric space. When is and when is not $A$ a complete metric space?

- Give examples of functions that satisfy and don't satisfy the Lipschitz condition.

- State and prove the Schwarz inequality.

- Suppose $(S_{1},d_{1})$ and $(S_{2},d_{2})$ are metric spaces. Prove that $(T,D)$ is a metric space, where $T=$ $S_{1}\times S_{2}$ and $D$ is the maximum of $d_1$ and $d_2$.

- State the fundamental lemma of differentiation. State and prove the chain rule for the composition of functions of two variables.

- State the Contraction Principle. State and prove the existence and uniqueness theorem for the initial value problem.

- Define the arc-length of a parametric curve and provide its basic properties. Provide the integral formula. Use it to find the arc-length of a circle.

- Define the curvature of a parametric curve. Find the curvature of the curve $(t^{2},t,5)$ as a function of $t>0.$ Under what circumstances is the acceleration perpendicular to the velocity?

- Suggest parametric equations for (a) circle in the plane, (b) an ascending spiral in space. Compare their curvatures based on the definition.