Midterm 2 Review

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Review questions for MIDTERM 2.

(a) Suppose $s(t)$ is the position (in feet) of an object moving along a line at time $t\geq 0$ seconds.

What is the average velocity between times $t=a$ and $t=b$ ?

$s(b)-s(a)$ $\frac {\text {ft}}{\text {second}}$ $\frac {s(b)-s(a)}{b-a}$ $\frac {\text {ft}}{\text {second}}$ $\frac {b-a}{s(b)-s(a)}$ $\frac {\text {ft}}{\text {second}}$ $b-a$ $\frac {\text {ft}}{\text {second}}$

(b) The table gives $s(t)$ over a three second interval.

$\begin {array}{|c|c|c|c|c|c|c|c|} \hline t& 0& 0.5& 1& 1.5& 2& 2.5& 3\\ \hline s(t)& 0 & 22 & 32 & 48 & 54 & 64 & 74 \\ \hline \end {array}$

The average velocity over the interval $[1,3]$ is $v_{av}=\answer {21}$ $\frac {\text {ft}}{\text {second}}$

32 $\frac {\text {ft}}{\text {second}}$ 10 $\frac {\text {ft}}{\text {second}}$ 26 $\frac {\text {ft}}{\text {second}}$ -5 $\frac {\text {ft}}{\text {second}}$ 48 $\frac {\text {ft}}{\text {second}}$

(a) Fill in the blanks: $f'(x)=\lim _{h\to \answer {0}}\frac {f(\answer {x+h})-f(\answer {x})}{\answer {h}}$

(b) Let $f(x)=\frac {1}{x-4}$ . Using the equation in part (a), compute:

$f'(x)=\lim _{h\to \answer {0}}\frac {\frac {1}{\answer {(x+h)-4}}-\frac {1}{\answer {x-4}}}{h}=\answer {-(x-4)^{-2}}$

The graph of a function $f$ is given below

(1) (a) Select the figure which has a secant line drawn through the points $(0,f(0))$ and $(4,f(4))$ .

(b) Select the figure which has the line tangent to the curve at $x=1$

(2) Choose the correct statement:

(i)

$f(1)=f(3)$ $f(1)>f(3)$ $f(1)<f(3)$

(ii)

$f'(1)=f'(3)$ $f'(1)>f'(3)$ $f'(1)<f'(3)$

(iii) Select the best approximation of $f'(3)$

$f'(3)\approx 2$ $f'(3)\approx -2$ $f'(3)\approx 1$ $f'(3)\approx -1$ $f'(3)\approx \frac {1}{2}$ $f'(3)\approx -\frac {1}{2}$

Let $f$ be a function such that $f(4)=3$ and $f'(4)=-6$ .

(a) Find the limit or say it does not exist (DNE).

$\lim _{x\to 4} \frac {f(x)-f(4)}{x-4}=\answer {-6}$

(b) An equation of the tangent line to the curve $y=f(x)$ at the point where $x=4$ is given by $y=\answer {-6x+27}$

(i) Given $e^{xy}=e+x-y,$ use implicit differentiation to find $\frac {dy}{dx}=\answer {\frac {1-ye^{xy}}{1+xe^{xy}}}$

(ii) Given y, select $\frac {dy}{dx}$ :

(a) $y=5\sin (x)\arctan \left (\frac {x}{x+1}\right )$

$5\cos (x)\frac {(x+1)^{-1}-x(x+1)^{-2}}{1+\frac {x^2}{(x+1)^2}}$ $5\sin (x)\frac {(x+1)^{-1}-x(x+1)^{-2}}{1+\frac {x^2}{(x+1)^2}}+5\cos (x)\arctan (\frac {x}{x+1})$ $5\sin (x)\frac {1}{1+\frac {x^2}{(x+1)^2}}+5\cos (x)\arctan (\frac {x}{x+1})$

(b) $y=\frac {1}{\sqrt {\tan (x^2)+5}}$

$\frac {-x\sec ^2(x^2)}{(\tan (x^2)+5)^{1.5}}$ $\frac {-1}{(\tan (x^2)+5)^{1.5}}$ $\frac {-x\sec ^2(2x)}{(\tan (x^2)+5)^{1.5}}$

$e^{ax}+axe^{ax}+a\arccos (ax)$ $ae^{ax}+\frac {a}{\sqrt {1-a^2x^2}}$ $e^{ax}+axe^{ax}+\frac {a}{\sqrt {1-a^2x^2}}$

(d) $y(x)=\left (\sin (x)\right )^{ax}$

$\sin (x)^{ax}(ax\cot (x)+a\ln (\sin (x)))$ $ax\sin (x)^{ax-1}\cos {x}$ $a\sin (x)^{ax}\ln (\sin (x))$

(ii) Find the following values or state the value is undefined (DNE):

(a) $\sin \left (\frac {17\pi }{6}\right )=\frac {1}{\answer {2}}$

(b) $\arcsin \left (\frac {17\pi }{6}\right )=\answer [format=string]{DNE}$

(d) $e^{-2\ln (3)}=\frac {1}{\answer {9}}$

(e) $\ln \left (-e^2\right )=\answer [format=string]{DNE}$

(f) $\ln \left (e^{2t}\right )=2\answer {t}$

(g) $10^{3\log _{10}(4)}=4^{\answer {3}}$

(h) $\ln (1)=\answer {0}$

(i) $\arctan \left (-\sqrt {3}\right )=\frac {\pi }{\answer {-3}}$

(j) $\left [\ddx \left (\cot (x)\right )\right ]_{\frac {\pi }{3}}=\answer {-\frac {4}{3}}$

(k) $\left [\ddx \left (\arctan (x)\right )\right ]_{-\sqrt {3}}=\answer {\frac {1}{4}}$

A table of values for $f(x)$ and $f'(x)$ , along with a graph of a function $g(x)$ is shown below. $\begin {array}{c|c|c} x & f(x)&f'(x)\\ \hline 1 & 2 & 4\\ \hline 2 & 3 & 5\\ \hline 3 & 4 & 1\\ \hline \end {array}$

(i) $\ddx \left [g(x)\right ]_{x=1}$ =

$\frac {13}{3}$ $\frac {4}{3}$ 12 4 DNE None of the above

(ii) $\ddx \left [f(g(x))\right ]_{x=3}$ =

-1 -4 $\frac {1}{4}$ 4 DNE None of the above

$\begin {array}{c|c|c} x & f(x)&f'(x)\\ \hline 1 & 2 & 4\\ \hline 2 & 3 & 5\\ \hline 3 & 4 & 1\\ \hline \end {array}$

(iii) $\ddx \left [g(f(x))\right ]_{x=2}$ =

-1 -5 $\frac {1}{5}$ 0 DNE None of the above

(iv) $f^{-1}(3)=$

4 1 $\frac {1}{3}$ 2 DNE None of the above

$\begin {array}{c|c|c} x & f(x)&f'(x)\\ \hline 1 & 2 & 4\\ \hline 2 & 3 & 5\\ \hline 3 & 4 & 1\\ \hline \end {array}$

(v) $\ddx \left [f^{-1}(x)\right ]_{x=3}$ =

$\frac {1}{4}$ $\frac {1}{5}$ 2 4 DNE None of the above

The function $f$ is defined by $f(x)=\frac {x}{\sqrt {x^2-9}}$ .

Answer the following question about the function $f$ .

(a) The domain of $f$ is $(\answer {-\infty },\answer {-3})\cup (\answer {3},\answer {\infty })$

(b) The function $f$ is

odd even neither

(d) $f''(x)=\answer {\frac {27x}{(x^2-9)^{2.5}}}$

(e) Select the correct statement for each interval:


On this interval	f is	f is
$(-\infty ,-3)$	inceasingdecreasingnot defined	concave upnot definedcocave down

$(-3,3)$	inceasingdecreasingnot defined	concave upnot definedcocave down

$(3,\infty )$	inceasingdecreasingnot defined	concave upnot definedcocave down

(f) Select the correct graph for $f$ :

(g) Is the function $f$ one-to-one:

Yes No

The graph of $f'$ (the derivative of $f$ ) on the interval $(-6,7)$ is shown in the figure.

Use the given graph of $f'$ to answer the following questions about f:

(a) On what interval(s) is $f$ decreasing?

ANSWER: $(\answer {-6},\answer {-2})$ and $(\answer {6},\answer {7})$

(b) List the x- coordinates of all critical points of $f$ (in ascending order).

ANSWER: $x=\answer {-2},\answer {4},\answer {6}$

ANSWER: $x=\answer {6}$

(d) List the x-coordinates of all critical points of $f$ that correspond to neither local maxima nor local minima?

ANSWER: $x=\answer {4}$

(e) On what intervals is $f$ concave up?

ANSWER: $(\answer {-6},0)$ and $(\answer {4},\answer {5})$

(f) List the x-coordinates of all inflection points of $f$ (in ascending order).

ANSWER $x=\answer {0},\answer {4},\answer {5}$

Given that $f(1)=2$ , $f'(1)=3$ , and $f''(1)=-1$ , find the following values or state ‘cannot be determined’ (CBD):

(a) $\left (\ddx \frac {(x+5)f'(x)}{f(x)}\right )_{x=1}=\answer {-15}$

(b) $\lim _{x\to 1}f(x)=\answer {2}$

(d) $\lim _{h\to 0}\frac {f(1+h)-f(1)}{h}=\answer {3}$

(e) $\lim _{h\to 0} \frac {f'(1+h)-f'(1)}{h}=\answer {-1}$

(f) $\lim _{x \to 1} \frac {f'(x)-f'(1)}{x-1}=\answer {-1}$

Simplify!
Given that $f(x)=\ln (\sec (x)+\tan (x))$ , $f'(x) = \answer {\sec (x)}$

The position, $s(t)$ , of an object moving along a horizontal line is given by $s(t)=3\sin \left (\frac {\pi }{4}t\right )$ , $t\geq 0$ , where $s$ is measured in feet and $t$ in seconds.

Complete the statements below.

(a) The position of the particle at time $t=2$ is $\answer {3}$ ft.

(b) The , $v_{av}$ , of the object over the interval $[0,t]$ is

$\answer {\frac {3\sin (\frac {\pi }{4}t)}{t}}$ $\frac {\text {ft}}{\text {s}}$ .

(c) Using the expression found in part (b), evaluate $\lim _{t\to 0^+} v_{av} = \answer {\frac {3\pi }{4}}$ $\frac {\text {ft}}{\text {s}}$

(e) What does the limit in part (c) represent?

The position at $t=0$ The acceleration at $t=0$ The instantaneous velocity at $t=0$

(f) Find the velocity of the particle: $v(t)=\answer {\frac {3\pi }{4}\cos (\frac {\pi }{4}t)}$ $\frac {\text {ft}}{\text {s}}$

(g) Find the acceleartion of the particle: $a(t)=\answer {-\frac {3\pi ^2}{16}\sin (\frac {\pi }{4}t)}$ $\frac {\text {ft}}{\text {s}^2}$

The figure below shows the graphs of $f$ , $f'$ , and another function $g$ .

Which curve is which: $f=\answer [format=string]{c}$ ; $f'=\answer [format=string]{a}$ ; $g=\answer [format=string]{b}$

The (entire) graph of a function $f$ is shown in the figure below.

Give answers for the following, or write ‘DNE’.

(a) Find the x-coordiantes of all points in the interval $(0,5)$ where $f$ has a local minimum.

ANSWER: $x=\answer {1}$

(b) Find the x-coordinate of all points in the interval $(0,5)$ where $f$ has a local maximum.

ANSWER: $x=\answer {2}$

ANSWER: $x=\answer {1},\answer {2}$

Assume that a function $f$ is continuous on its domain, $(-1,6)$ . The graph of $f'$ , the derivative of $f$ , is shown in the figure below.

(a) Write the x-coordinates of all critical points of $f$ (or write NONE), in ascending order.

ANSWER: $x=\answer {0},\answer {3},\answer {5}$

(b) Write the x-coordinates of all local maxima of $f$ (or write NONE).

ANSWER: $x=\answer {5}$

ANSWER: $x=\answer {3}$

(d)Find the interval(s) on which $f$ is increasing.

ANSWER: $(\answer {3},\answer {5})$

(e) Find the interval(s) on which $f$ is concave down.

ANSWER: $(\answer {0},\answer {1})$

(f) Write the x-coordinates of all inflection points (or write NONE), in ascending order.

ANSWER: $x=\answer {0},\answer {1}$

A curve is given by the equation $\tan (3y+x)=x^2-1$ .

(i) (a) Use implicit differentiation to find the derivative.

$\frac {dy}{dx}=\answer {\frac {2x}{3\sec ^2(3y+x)}-\frac {1}{3}}$

(b) Check (algebraically), that the point $(1,-\frac {1}{3})$ lies on the curve.

ANSWER: $\left [\frac {dy}{dx}\right ]_{x=1, y=-\frac {1}{3}}=\answer {\frac {1}{3}}$

(ii) Part of the curve $\tan (3y+x)=x^2-1$ that contains the point $(1,-\frac {1}{3})$ is shown in the figure below:

(a) Find an explicit expression for $y$ , $y=y(x)$ , represented by the graph above.

$y(x)=\answer {\frac {1}{3}(\arctan (x^2-1)-x)}$

(b) Using the explicit expression in part (a), find $\frac {dy}{dx}$ .

$\frac {dy}{dx}=\answer {\frac {1}{3}(\frac {2x}{(x^2-1)^2+1}-1)}$

(d) Write an equation of the tangent line to the graph of $y=y(x)$ at the point where $x=1$ .

ANSWER: $y=\answer {\frac {1}{3}(x-1)-\frac {1}{3}}$

A function $f'$ (the derivative of $f$ ) is given by $f'(x)=(x-5)^3$ . Answer the following, or write ”DNE” (does not exist).

(a) List all interval(s) on which $f$ is increasing.

ANSWER: $(\answer {5},\answer {\infty })$

(b) List x-coordinates of all points where $f$ has a local maximum.

ANSWER: $x=\answer [format=string]{DNE}$

ANSWER: $x=\answer {5}$

(d) Find $f''(x)$ .

ANSWER: $f''(x)=\answer {3(x-5)^2}$

(e) List all interval(s) on which $f$ is concave up.

ANSWER: $(\answer {-\infty },\answer {\infty })$

(f) List x-coordinates of all inflection points of $f$ .

ANSWER: $x=\answer [format=string]{DNE}$

The function $s(t)=\frac {12}{t+1}$ , measured in meters, gives the position of an object moving along a line at time $t\geq 0$ , measured in seconds.

(a) Is the function $s$ one-to-one?

Yes No

(b) Find the expression for inverse of $s$ , $s^{-1}(t)$ .

ANSWER: $s^{-1}(t)=\answer {\frac {12}{t}-1}$

ANSWER: $s^{-1}(5) =\answer {\frac {7}{5}}$

(d) Interpret your answer to part (c), by selecting the correct answer.

$s^{-1}(5)$ gives the speed of the object at time 5 $s^{-1}(5)$ gives the position of the object at time 5 $s^{-1}(5)$ gives the time when the object was at position 5 $s^{-1}(5)$ gives the time when the object had speed 5

(e) Find the average velocity between times $t=0$ and $t=2$ .

ANSWER: $v_{av}=\answer {-4}$ $\frac {\text {m}}{\text {s}}$

(f) Find an expression for the instantaneous velocity at time $t>0$ .

ANSWER: $v(t)=\answer {-12(t+1)^{-2}}$ $\frac {\text {m}}{\text {s}}$

(g) Find an expression for the instantaneous acceleration at time $t>0$ .

ANSWER: $a(t)=\answer {24(t+1)^{-3}}$ $\frac {\text {m}}{\text {s}^2}$

(h) Is the velocity increasing or decreasing for $t>0$ ? Select the correct answer.

Increasing Decreasing

(i) Is the speed increasing or decreasing for $t>0$ ? Select the correct answer.

Increasing Decreasing

A ladder $10$ ft long rests against a vertical wall. If the bottom of the ladder slides away from the wal at a speed of $2$ ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac {\pi }{3}$ radians? $\answer {\frac {2}{5}}$ $\frac {\text {rad}}{\text {s}}$

The position function (in feet) for an object at time $t$ (in minutes), $0\leq t\leq 10$ , is given by $s(t)=-40t^2+160t+480$ .

(a) Find the velocity $v(t)$ at any time $t$ .

ANSWER: $v(t)=\answer {-80t+160}$ $\frac {\text {ft}}{\text {min}}$

(b) Find the acceleration $a(t)$ at any time $t$ .

ANSWER: $a(t)=\answer {-80}$ $\frac {\text {ft}}{\text {min}^2}$

ANSWER: At the time $t=\answer {2}$ min.

(d) What are the velocity and acceleration at that time?

ANSWER: $v=\answer {0}$ $\frac {\text {ft}}{\text {min}}$ , $a=\answer {-80} \frac {\text {ft}}{\text {min}^2}$

(e) At what (positive) time is the object at the origin?

ANSWER: At the time $t=\answer {6}$ min.

(f) What are the velocity and acceleration at that time?

ANSWER: $v=\answer {-320}$ $\frac {\text {ft}}{\text {min}}$ , $a=\answer {-80}$ $\frac {\text {ft}}{\text {min}^2}$

(g) Find the time interval(s) when the velocity is decreasing.

ANSWER: $(\answer {0},\answer {10})$

The (entire) graph of a function $f$ is given in the figure below:

(a) Find the value.
$\ddx \left [f^{-1}(x)\right ]_{x=-3}=\answer {-2}$

(b) List the x-coordinates of all critical points of $f$ .

ANSWER: $x=\answer {3}$

(d) List the x-coordinates of all local maxima in of $f$ , or say ‘none’.

$x=\answer [format=string]{none}$

(g) Select the correct graph of $f'$ .

(i) Let $f(x)=x^4-8x^2+10$ .

List the x-coordinates of all critical points of $f$ in ascending order.

ANSWER: $x=\answer {-2},\answer {0},\answer {2}$

(ii) Let $f(x)=x\ln (x)$

(a) Find the domain of the function $f$ .

ANSWER: $(\answer {0},\answer {\infty })$

(b) List the x-coordinates of all critical points of $f$ in ascending order.

ANSWER: $x=\answer {\frac {1}{e}}$

(iii) Let $f(x)=x\sqrt {6-x^2}$

(a) Find the domain of the function $f$ .

ANSWER: $[\answer {-\sqrt {6}},\answer {\sqrt {6}}]$

(b) State the interval(s) of continuity of $f$ .

ANSWER: $[\answer {-\sqrt {6}},\answer {\sqrt {6}}]$

ANSWER: $\answer {-\sqrt {3}},\answer {\sqrt {3}}$

A water tank is to be drained for cleaning. There are $V$ liters of water left in the tank $t$ minutes after the draining began, where $V=42(60-t)^2$

(a) Find the average rate at which water drains during the first $10$ minutes.

ANSWER: $\answer {-4620}$ $\frac {\text {liters}}{\text {min}}$

(b) Find the rate at which the volume of water is changing $10$ minutes after the draining began.

ANSWER: $\answer {-4200}$ $\frac {\text {liters}}{\text {min}}$

$\frac {d}{dt}\left (\frac {\frac {dV}{dt}(t)}{V(t)}\right )=\answer {-2(60-t)^{-2}}$

(d) What are the units of the answer to part (c)?

(e) Find the rate of the rate at which the volume of water is changing 10 minutes after draining begins.

ANSWER: $\answer {84}$ $\frac {\text {liters}}{\text {min}^2}$

(f) Is the rate at which the volume of the water is changing increasing or decreasing (during the draining)?

Increasing Decreasing

(g) Assume that the tank has the shape of a rectangular box $7$ m long, $6$ m wide, and $5$ m high. What is the rate of change of the water depth when the water depth is $3$ m? (HINT: 1 liter = .001 $m^3$ )

ANSWER: $\answer {\frac {-\sqrt {30}}{50}}$ $\frac {\text {m}}{\text {min}}$

We are inflating a spherical balloon at a rate of 3 $cm^3/sec$ .

(a) At what rate is the radius increasing when the radius is $6$ cm?

ANSWER: $\answer {\frac {1}{48\pi }}$ $\frac {\text {cm}}{\text {sec}}$

(b) At what rate is the surface area increasing at that moment?

ANSWER: $\answer {1}$ $\frac {\text {cm}^2}{\text {sec}}$

Two cars leave an intersection. One heads west at $30$ mi/hr. The other leaves 30 minutes later, heading north at $40$ mi/hr. How fast is the distance between them changing $1$ hr after the first car left the intersection?

ANSWER: $\answer {\frac {170}{\sqrt {13}}}$ $\frac {\text {mi}}{\text {hr}}$