APEX Calculus

1.

True or False? Given parametric equations $x=f(t)$ and $y=g(t)\text{,}$ $\lz{y}{x} = \fp(t)/g'(t)\text{,}$ as long as $g'(t) \neq 0\text{.}$

• True

• False

2.

Given parametric equations $x=f(t)$ and $y=g(t)\text{,}$ the derivative $\frac{dy}{dx}$ as given in Key Idea 9.3.2 is a function of ?

3.

True or False? Given parametric equations $x=f(t)$ and $y=g(t)\text{,}$ to find $\lzn{2}{y}{x}\text{,}$ one simply computes $\lzoo{t}{\lz{y}{x}}\text{.}$

• True

• False

4.

True or False? If $\lz{y}{x}=0$ at $t=t_0\text{,}$ then the normal line to the curve at $t=t_0$ is a vertical line.

• True

• False

5.

$x=t\text{,}$ $y=t^2\text{;}$$t=1$

6.

$x=\sqrt{t}\text{,}$ $y=5t+2\text{;}$$t=4$

7.

$x=t^2-t\text{,}$ $y=t^2+t\text{;}$$t=1$

8.

$x=t^2-1\text{,}$ $y=t^3-t\text{;}$$t=0$ and $t=1$

9.

$x=\sec(t)\text{,}$ $y=\tan(t)$ on $(-\pi/2,\pi/2)\text{;}$$t=\pi/4$

10.

$x=\cos(t)\text{,}$ $y=\sin(2t)$ on $[0,2\pi]\text{;}$$t=\pi/4$

11.

$x=\cos(t) \sin(2t)\text{,}$ $y=\sin(t) \sin(2t)$ on $[0,2\pi]\text{;}$ $t=3\pi/4$

12.

$x=e^{t/10}\cos(t)\text{,}$ $y=e^{t/10}\sin(t)\text{;}$ $t=\pi/2$

13.

$x=t\text{,}$ $y=t^2$

14.

$x=\sqrt{t}\text{,}$ $y=5t+2$

15.

Find the values of $t$ where the curve defined by $x=t^2-t\text{,}$ $y=t^2+t$ has a horizontal tangent line. (If there is more than one, use commas to separate them.)

$t\in\big\{$$\big\}$

At what point(s) in the plane are the horizontal tangent lines? (If there is more than one, use commas to separate them.)

16.

$x=t^2-1\text{,}$ $y=t^3-t$

17.

$x=\sec(t)\text{,}$ $y=\tan(t)$ on $(-\pi/2,\pi/2)$

18.

Find the values of $t$ in $[0,2\pi)$ where the curve defined by $x=\cos(t)\text{,}$ $y=\sin(2t)$ has a horizontal tangent line. (If there is more than one, use commas to separate them.)

$t\in\big\{$$\big\}$

At what point(s) in the plane are the horizontal tangent lines? (If there is more than one, use commas to separate them.)

19.

$x=\cos(t) \sin(2t)\text{,}$ $y=\sin(t) \sin(2t)$ on $[0,2\pi]$

20.

$x=e^{t/10}\cos(t)\text{,}$ $y=e^{t/10}\sin(t)$

21.

Find $t=t_0$ where the graph of $x=\frac{1}{t^2+1}\text{,}$ $y=t^3$ is not smooth, then find $\lim\limits_{t\to t_0}\lz{y}{x}\text{.}$

$t_0=$

$\lim\limits_{t\to t_0}\lz{y}{x}=$

22.

Find $t=t_0$ where the graph of $x=-t^3+7t^2-16t+13\text{,}$ $y=t^3-5t^2+8t-2$ is not smooth, then find $\lim\limits_{t\to t_0}\lz{y}{x}\text{.}$

$t_0=$

$\lim\limits_{t\to t_0}\lz{y}{x}=$

23.

$x=t^3-3t^2+3t-1\text{,}$$y=t^2-2t+1$

24.

$\ds x=\cos^2(t)\text{,}$$y=1-\sin^2(t)$

25.

$x=t\text{,}$$y=t^2$

26.

$x=\sqrt{t}\text{,}$$y=5t+2$

27.

For the curve defined by $x=t^2-t\text{,}$ $y=t^2+t\text{,}$ find $\lzn{2}{y}{x}$ (as a function of $t$). Then determine the intervals on which the graph of the curve is concave up/down.

$\lzn{2}{y}{x}=$

The curve is concave up for $t$ in the interval(s) . (If there is more than one interval, use commas to separate them.)

The curve is concave down for $t$ in the interval(s) . (If there is more than one interval, use commas to separate them.)

28.

$x=t^2-1\text{,}$$y=t^3-t$

29.

$x=\sec(t)\text{,}$$y=\tan(t)$ on $(-\pi/2,\pi/2)$

30.

For the curve defined by $x=\cos(t)\text{,}$ $y=\sin(2t)\text{,}$ find $\lzn{2}{y}{x}$ (as a function of $t$). Then determine the intervals (within $[0,2\pi)$) on which the graph of the curve is concave up/down.

$\lzn{2}{y}{x}=$

The curve is concave up for $t$ in the interval(s) . (If there is more than one interval, use commas to separate them.)

The curve is concave down for $t$ in the interval(s) . (If there is more than one interval, use commas to separate them.)

31.

$\ds x=\cos(t) \sin(2t)\text{,}$$y=\sin(t) \sin(2t)$ on $[-\pi/2,\pi/2]$

32.

$x=e^{t/10}\cos(t)\text{,}$$y=e^{t/10}\sin(t)$

33.

Find the arc length of the graph defined by $x=-3\sin(2t)\text{,}$ $y=3\cos(2t)$ on $[0,\pi]\text{.}$

34.

Find the arc length of the graph defined by $x=e^{t/10}\cos(t)\text{,}$ $y=e^{t/10}\sin(t)$ on $[0,2\pi]$ and $[2\pi,4\pi]\text{.}$

On $[0,2\pi]\text{,}$ the length is .

On $[2\pi,4\pi]\text{,}$ the length is .

35.

Find the arc length of the graph defined by $x=5t+2\text{,}$ $y=1-3t$ on $[-1,1]\text{.}$

36.

$x=2t^{3/2}\text{,}$$y=3t$ on $[0,1]$

37.

Approximate the arc length of one petal of the rose curve $x=\cos(t) \cos(2t)\text{,}$$y=\sin(t) \cos(2t)$ using Simpson's Rule and $n=4\text{.}$

38.

Approximate the arc length of the “bow tie curve” $x=\cos(t)\text{,}$$y=\sin(2t)$ using Simpson's Rule and $n=6\text{.}$

39.

Approximate the arc length of the parabola $x=t^2-t\text{,}$$y=t^2+t$ on $[-1,1]$ using Simpson's Rule and $n=4\text{.}$

40.

A common approximate of the circumference of an ellipse given by $x=a\cos(t)\text{,}$$y=b\sin(t)$ is $\ds C\approx 2\pi\sqrt{\frac{a^2+b^2}2}\text{.}$ Use this formula to approximate the circumference of $x=5\cos(t)\text{,}$ $y=3\sin(t)$ and compare this to the approximation given by Simpson's Rule and $n=6\text{.}$

41.

Find the surface area of the sphere formed by rotating the circle $x=2\cos(t)\text{,}$$y=2\sin(t)$ about:

1. the $x$-axis and

2. the $y$-axis.

42.

Find the surface area of the torus (or “donut”) formed by rotating the circle $x=\cos(t) +2\text{,}$$y=\sin(t)$ about the $y$-axis.

43.

Approximate the surface area of the solid formed by rotating the “upper right half” of the bow tie curve $x=\cos(t)\text{,}$$y=\sin(2t)$ on $[0,\pi/2]$ about the $x$-axis, using Simpson's Rule and $n=4\text{.}$

44.

Approximate the surface area of the solid formed by rotating the one petal of the rose curve $x=\cos(t) \cos(2t)\text{,}$$y=\sin(t) \cos(2t)$ on $[0,\pi/4]$ about the $x$-axis, using Simpson's Rule and $n=4\text{.}$