Spring 2017 Exam 2

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Here you can work on a practice exam 2. This exam was administered in Spring 2017.

Math 160 Calculus for Physical Scientists I
Exam 2 - Version 1
March 9, 2017, 5:00-6:50 pm

If the radius of a circle increases from $r=A$ to $r=B$ , then the average rate of change of the area of the circle is

$\pi (B^2-A^2)$ $\pi (B-A)$ $\displaystyle \pi \left (B^2+A^2\right )$ $\displaystyle \pi \left (B+A\right )$ None of the above

We know that $f(2)=4$ and $f'(2)=3$ , then the value of $\displaystyle \frac {d}{dx}\left (\frac {3x^2}{f(x)}\right )\bigg |_{x=2}$

$\displaystyle \frac {21}{4}$ $\displaystyle \frac {4}{3}$ $\displaystyle \frac {3}{4}$ 4 None of the above.

If $h'(t)$ is continuous on $(-\infty ,\infty )$ , then

$\displaystyle \lim _{t\rightarrow 100}h(t)=h'(100)$ . $\displaystyle \lim _{t\rightarrow 100} h(t)= h(100)$ . $\displaystyle \lim _{t\rightarrow 100}h(t)$ may not exist. $\displaystyle \lim _{h\rightarrow 0}\frac {h(100+h)-h(100)}{h}$ may not exist. None of the above.

The function $f(t)$ represents the distance of a vehicle moving along a straight road from its starting point at time $t$ .

Below are three graphs of the derivative, $f'(t)$ , of the distance function. Which graph best matches each of the following vehicle scenarios?

Graph 1:

Graph 2:

Graph 3:

Scenario 1: A car in heavy traffic conditions.

Graph 1 Graph 2 Graph 3

Scenario 2: A bus on a popular route, with no traffic.

Graph 1 Graph 2 Graph 3

Scenario 3: A car with no traffic and all green lights.

Graph 1 Graph 2 Graph 3

Below is the graph of the function, $f(x)$ .

Of all the properties listed below, choose all properties that are reflected in the graph of $f(x)$ .

$f(x)$ is not continuous at $x=-4$ $f(x)$ is not continuous at $x=-1$ $f(x)$ is not continuous at $x=0$ $f(x)$ is not differentiable at $x=-4$ $f(x)$ is not differentiable at $x=-1$ $f(x)$ is not differentiable at $x=0$ $f'(x)=0$ at $x=-2$ $f'(x)=0$ at $x=0$ $f'(x)<0$ for $-4<x<-2$ $f'(x)<0$ for $-2<x<-1$ $f'(x)<0$ for $-1<x<0$ $f'(x)<0$ for $x>0$ $f'(x)>0$ for $-4<x<-2$ $f'(x)>0$ for $-2<x<-1$ $f'(x)>0$ for $-1<x<0$ $f'(x)>0$ for $x>0$ $y=4$ is an absolute maximum $y=3$ is an absolute maximum $y=-0.5$ is an absolute minimum $y=-1.5$ is an absolute minimum

Use $g(t)=\tan (f(t))$ to answer the following. When referring to $f'(t)$ , use $d(t)$ .

Find the derivative of $g(t)$ .

$\answer [given]{(\sec (f(t)))^2 * d(t)}$

Using your result from part (a) and the information provided in the table below, find the value of $g'(\pi )$ . Circle the correct answer.

$\begin {array}{c||c|c} \, & f(t) & f'(t)\\ \hline \hline \, & \, & \, \\ t=0 & 2 & -\frac {1}{2}\\ \, & \, & \, \\ \hline \, & \, & \, \\ t=\pi & \frac {\pi }{3} & \pi \\ \end {array}$

$g'(\pi )=4\pi$ $g'(\pi )=4$ $g'(\pi )=\pi$ $g'(\pi )=0$ $g'(\pi )=1$

Suppose that $f(t)=3t$ , then $g(t)=\tan (3t)$ .

What is $g''(t)$ ? Circle the correct answer.

$3\sec ^2(3t)$ $18\sec ^2(3t)\tan (3t)$ $2\sec ^2(3t)\tan (3t)$ $6\sec ^2(3t)\tan (3t)$ $2\sec (3t)$

Below is the graph of the implicitly defined function $\displaystyle \sin \left (\pi y\right )+3y^2=3x^3$ .

It can be seen in the picture that the point $(1,-1)$ is on the graph of $\displaystyle \sin \left (\pi y\right )+3y^2=3x^3$ . Algebraically verify this fact. (Note that your response cannot be validated as correct or incorrect in this system).

Use calculus to find $\displaystyle \frac {dy}{dx}$ .

$\displaystyle \dfrac {dy}{dx} = \answer [given]{\dfrac {9x^2}{\pi *\cos (\pi *y) + 6y}}$

Find the slope of the line tangent to $\displaystyle \sin \left (\pi y\right )+3y^2=3x^3$ at the point $(1,-1)$ .

The slope of the line is $\answer [given]{\dfrac {9}{-1(\pi +6)}}$ .

Find the equation of the line tangent to $\displaystyle \sin \left (\pi y\right )+3y^2=3x^3$ at the point $(1,-1)$ .

In point-slope form, the equation of the line tangent to the curve at $(1,-1)$ is

$y - \answer [given]{-1} = \answer [given]{\dfrac {9}{-1(\pi +6)}} ( x - \answer [given]{1}).$

Indicate whether each of the following statements is True or False.

If the statement is true, explain how you know it’s true.

If it is false, give a counterexample and explain why it is a counterexample. (A counterexample is an example of a function for which the ‘‘if’’ part of the statement is true, but the ‘‘then’’ part is false.) A graph with an explanation can be used as a counterexample.

If $f$ is continuous at $x=1$ , then $f$ is differentiable at $x=1$

True False

Given that $f(x)$ is continuous on the interval $(1,3)$ , then $f(x)$ attains an absolute maximum on the interval $(1,3)$ .

True False

Two different functions, $f(x)$ and $g(x)$ , cannot have the same derivative functions unless both $f(x)$ and $g(x)$ are linear functions with the same slope.

True False

Consider the function $f(x)$ given by

$f(x) = \begin {cases} \displaystyle -x^3, & x \leq 0 \\ x^3, & x >0 \end {cases}$

Use the definition of the derivative of a function at a point (as a limit) to determine whether $f(x)$ is differentiable at $x=0$ . Show details of how you made this determination. Evaluate any limits involved algebraically (without using a calculator).

If $f'(0)$ exists, fill in the blank with its value. If not, draw a smiley face in the blank. $\displaystyle \lim _{h\to 0^-} \answer [given]{\dfrac {-h^3}{h}}$

$\displaystyle \lim _{h\to 0^+} \answer [given]{\dfrac {h^3}{h}}$

$f'(0) = \answer [given]{0}$

Sketch a graph of $f(x)$ on the grid provided.
(Note that this question cannot be validated in this system.)

Describe how your graph above supports the answer you found in the first question of this problem.
(Note that this question cannot be validated in this system.)

There are three graphs plotted in the coordinate system below ( $f(x)$ , $f'(x)$ , and $f''(x)$ ).

Which graph is $f(x)$ ? Which graph is $f'(x)$ ? Which graph is $f''(x)$ ? Give reasons for your answers in sentences. Your explanation should include a discussion of slope with regard to each graph.

Graph 1 is

$f(x)$ $f'(x)$ $f''(x)$

Graph 2 is

$f(x)$ $f'(x)$ $f''(x)$

Graph 3 is

$f(x)$ $f'(x)$ $f''(x)$

Justify your answers using complete sentences. Your justification should include a discussion of slope with regard to each graph.
(Note that the response below cannot be validated as correct or incorrect.)

Calculate the indicated derivatives by using the Differentiation Rules (Theorems). Answers must be accompanied by supporting work that shows how you calculated the derivative. You do not need to simplify your answers on these problems. If you do simplify an answer, you must simplify correctly.

$\displaystyle \frac {d}{dx}\left (\frac {3}{x^3}+\cos (x)+\pi ^4\right )$ $= \answer {\dfrac {-9}{x^4} - \sin (x)}$

$\displaystyle \frac {d}{d\theta }\left [\frac {5\theta }{\sin (\theta )+2}\right ]$
(Instead of using $\theta$ in your answer, use $t$ in the answer field below.)
$= \answer {\dfrac {5*\sin (t) + 10 - 5*t*\cos (t)}{(\sin (t) + 2)^2}}$

Find $g'(t)$ given $\displaystyle g(t)=(3t-t^{4/3})(4\sqrt {t}+2t^3)$ $= \answer {12t^(3/2) + 6t^4 - 4t^(11/6) - 2t^(13/3)}$