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Calculus 1: exercises

This is a set of exercises for Calculus 1: course.

• Let $h(x)=x²+3x-10$. Find the $x$- and $y$-intercepts and sketch the graph of the function.
• Sketch the graph of a function f that has the following property: f is increasing and concave down on $[-1,1]$.
• Find the formulas (only) of the inverses of the following functions: 1. $f(x)=(x+1)³$. 2. $g(x)=\ln (x³)$.
• $f′(x)=(e^{x})².$ (a) On what intervals, if any, if f increasing? (b) On which intervals, if any, is f concave down? (c) Sketch the graph of f.
• Find the derivatives of the functions: (a) $3x^{e}+e^{π}$ (b) $7\ln x+(1/x)-\ln 2$
• Find antiderivatives of the functions: (a) $2\sqrt{x}+3$ (b) $5e^{x}+17x$
• Suppose that F is an antiderivative of a differentiable function $f$. If F is increasing on $[a,b]$, what is true about $f$?
• Evaluate $∫x\cos (1-x²)dx$
• A ladder, $12$ feet long, is leaning against a wall. If the foot of the ladder slides away from the wall along level ground, what is the rate of change of the top of the ladder with respect to the distance of the foot of the ladder from the wall when the foot is 6 ft from the wall?
• Is infinity a limit?
• Is infinity a number?

1 Analytic geometry

Indicate if the following statements are true or false.

1. A parabola does not intersect its directrix.
2. The center of an ellipse is the midpoint between its foci.
3. For a hyperbola, the rays of light parallel to the axis are reflected to the focus.
4. The equation $x²/a²+y²/b²=-1$ represents an ellipse.
5. There is an ellipse with the focal length equal to zero.

2 Polar coordinates

Indicate if the following statements are true or false:

1. In polar coordinates, $(1,π/2)$ and $(-1,-π/2)$ represent the same point.
2. The curve $r=3+\cos θ$ passes through the origin.
3. The curve $r=\cos 2θ$ is closed.
4. The curve $r=1+\cos θ$ is bounded.
5. The graph of $r=θ²$ can be represented as a parametric curve in Cartesian coordinates.
6. In polar coordinates, $A=(1,π/2)$ and $B=(-1,-π/2)$ represent the same point.
7. The slope of the polar curve $r=0$ is equal to $0$.
8. The graph of the curve $r=\cos 2θ$ is is a spiral.
9. The parametric curve $x=t²,y=\sin t$ is bounded.
10. The graph of $r=θ²$ can be represented as a parametric curve in Cartesian coordinates.

3 Integration

Complete the following identities:

1. $(f(x)⋅x²)′=f′(x)⋅x²+...$
2. $∫x⁻¹dx=...$
3. $∫f′(x)dx=...$
4. $∫udv=uv...$
5. $u=\cos t => du=...$