Midterm 3 Review

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Review questions for MIDTERM 3.

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?

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

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

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

$x=\answer {6}$

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

$x=\answer {4}$

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

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

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

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

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.

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

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

$x=\answer {5}$

$x=\answer {3}$

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

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

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

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

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

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

Draw a possible graph of $f$ given that it satisfies all of the following conditions.

(a) Domain of $f=(-\infty ,-2)\cup (-2,\infty )$ ,

(b) f is continuous on its domain and differentiable all all points in the domain except at $x=6$

(d) $\lim _{x\to -2} f(x)=-8$ , $\lim _{x\to -\infty }f(x)=1$ , $\lim _{x\to \infty }f(x)=1$ ,

(e) $f'(x)<0$ on $(-\infty ,-2)$ , and on $(6,\infty )$ ,

(f) $f'(x)>0$ on $(-2,6)$ ,

(g) $f''(x)<0$ on $(-\infty ,-2)$ and on $(-2,2)$ ,

(h) $f''(x)>0$ on $(2,6)$ and on $(6,\infty )$

Once you’ve finished, select ‘Done’, and compare your answer with the one shown

Done

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.
$(\answer {5},\answer {\infty })$

(b) List x-coordinates of all points where $f$ has a local maximum or write DNE.
$x=\answer [format=string]{DNE}$

$x=\answer {5}$

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

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

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

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

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

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

Evaluate the following limits. You may use L‘Hospital’s Rule

(a) $\lim _{x\to 0^+} (e^x-1)^\frac {1}{x}=\answer {0}$

(b) $\lim _{x\to 0^+} \tan (x)^{x^2}=\answer {1}$

(d) $\lim _{x\to 0} \frac {e^x-1-x}{x^2}=\answer {\frac {1}{2}}$

(e) $\lim _{x\to (\frac {\pi }{2})^-} \frac {\cos (x)\sin (2x)}{(x-\frac {\pi }{2})^2}=\answer {2}$

(f) $\lim _{x\to \infty } (x-\sqrt {x^2+4x})=\answer {-2}$

The figure shows a right triangle in the first quadrant. One side of the triangle is on the x-axis; its hypotenuse runs from the origin to a point on the parabola $y=4-x^2$ . Find the coordinates $x$ and $y$ that maximize the area of the triangle.

$x=\answer {\frac {2}{\sqrt {3}}}, y=\answer {\frac {8}{3}}$

(i) The point on the curve $y=\sqrt {x}$ that is closest to the point $(4,0)$ occurs at $x=\answer {\frac {7}{2}}$

(ii) A cruise line offers a trip for $1000 per passenger. If at least 100 passengers sign up, the price is reduced for all passengers by $5 for every additional passenger (beyond 100) who goes on the trip. The boat can accommodate 250 passengers.

The number of passengers which maximizes the cruise line’s total revenue is $\answer {150}$

What price does each passenger pay if that number of passengers goes on the cruise? $\answer {750}$

(iii) Find the dimensions of the right circular cylinder of maximum volume that can be placed inside a sphere of radius R: $r=\answer {R\sqrt {\frac {2}{3}}}, h=\answer {\frac {2R}{\sqrt {3}}}$

(iv) A certain tank consists of a right circular cylinder with hemispherical ends. For a given surface area S, find the dimensions (radius and length) of the tank with maximum volume (your answer should include S): $r=\answer {\sqrt {\frac {S}{4\pi }}}, l=\answer {0}$

(v) A square piece of tin 24 in on each side is to be made into an open-top box by cutting a small square from each corner and bending up the flaps to form the sides. What is the side length of the square that should be cut from each corner to make the volume of the box as large as possible? $s=\answer {4}$

A toy roller-coaster has been designed so that the rail has the shape of the curve given in the figure below, where $f(x)=x-\sin (\frac {\pi }{5}x)$ (x in inches, $f(x)$ gives the altitude)

The average rate of change of the altitude of the roller coaster on the interval $[0,10]$ is $\answer {1}$ .

Select the best interpretation of $f'(a)$ for $0<a<10$ .

$f'(a)$ is the altitude at $a$ . $f'(a)$ is the average rate of change of the altitude on the interval $[0,a]$ . $f'(a)$ is the velocity of an object at $a$ . $f'(a)$ is the instantaneous rate of change of the altitude at $a$ . $f'(a)$ is the slope of the secant line passing through $(0,0)$ and $(10,10)$ .

Because $f$ is on the interval $[0,10]$ and $f$ is on the interval $(0,10)$ , $f$ satisfies the conditions of the Mean Value Theorem.

By the Mean Value Theorem, there exists $c$ in $(0,10)$ such that $f'(c) = 1$ . In face this happens twice, when $c_1=\answer {\frac {5}{2}}$ and when $c_2=\answer {\frac {15}{2}}$ (assume $c_1<c_2$ ).

The steepest point on the roller coaster is $\left (\answer {5},\answer {5}\right )$ . (Hint: maximize $f'(x)$ on $(0,10)$ .)

The linear approximation, $L$ , to $f$ at $a=5$ is $L(x) = \answer {\left (1+\frac {\pi }{5}\right )(x-5) + 5}.$

Using this linear approximation we estimate that $f(7)$ is approximately $f(7) \approx \answer {\left (1+\frac {\pi }{5}\right )2+5}.$ This estimate is an because $f$ is concave upconcave down between 5 and 7.

When $x$ changes from $x=5$ to $x+\Delta x=7$ the change in $f$ , $\Delta y$ is $\Delta y = \answer {2-\sin \left (\frac {7\pi }{5}\right )}.$ We can approximate this change with the differential $\mathrm {d}y$ which is $\mathrm {d}y = \answer {\left (1+\frac {\pi }{5}\right )2}.$

Select all correct answers for each question below.

(i) At what point(s) $c$ does the conclusion of the Mean Value Theorem hold for $f(x)=x^2$ on the interval $[0,2]$ ?

0 1 2 $\frac {1}{2}$ Such a point does not exit None of the above

(ii) The equation of the line that represents the linear approximation to the function $f(x)=\ln (x)$ at $a=1$ is

$y=x$ $y=-x$ $y=ln(x-1)$ $y=x-1$ Such a line does not exit None of the above

(iii) Determine the following indefinite integral $\int (\sec ^2(x)+5)dx$

$\frac {\sec ^3(x)}{3}+5x+C$ $2\sec (x)\tan (x)+5x+C$ $2\sec ^2(x)\tan (x)+C$ $\tan (x)+C$ $\tan (x)+5x+C$ None of the above

(iv) Evaluate the expression $\sum _{k=1}^3(4k-1)$

$11$ $21$ $4k-1$ $3$ None of the above

(i) The population of a culture of cells grows according to the function $P(t)$ , where $t\geq 0$ is measured in weeks.

$\begin {array}{|c|c|}\hline t & P(t) \\ \hline 0 & 0 \\ \hline 1 & 40 \\ \hline 3 & 65 \\ \hline 4 & 80 \\ \hline \end {array}$

(a) What is the average rate of change in the population during the time interval $[0,4]$ ? $\answer {20}$ $\frac {\text {cells}}{\text {week}}$

(b) Assume that $P(t)$ is continuous on $[0,+\infty )$ and differentiable on $(0,+\infty )$ . Is there a moment in which the instantaneous rate of change of $P$ is equal to the average rate of change computed in part (a)?

YesNo

If yes, select the theorem which guarantees the existence of such a point. If no, select ‘No Theorem’.

Intermediate Value TheoremMean Value TheoremL‘Hospital’s RuleNo Theorem

(ii) Let $f(x)=\frac {4}{x}$ . The graph of $f$ is given in the figure below.

(a) The linear approximation L to the function $f$ at $a=2$ is $L(x)=\answer {4-x}$

(b) Select the figure which includes the graph of $L$ :

(d) The approximation in part (c) is an because $f$ is concave upconcave down .

Given $f(x)=8x-x^2$ on $[0,8]$ , and $n=4$ , complete the following steps:

(a) Calculate $\Delta x=\answer {2}$ and the grid points (in ascending order) $x_0=\answer {0}, x_1=\answer {2}, x_2=\answer {4}, x_3=\answer {6}, x_4=\answer {8}$

(b) Select the figure with the right Riemann sum drawn in

The figure illustrates the Riemann sum $\sum _{k=1}^{n}f(x_k^*)\Delta x$ for $f$ on the interval $[4,8]$ . Use the figure to answer the following questions.

(i) Given the interval $[4,8]$ , what is $n$ ?

n=3 n=8 n=5 n=4 None of the above

(ii) What is $\Delta x$ ?

4 0 1 2 $\frac {4}{k}$ None of the above

(iii) Given that $x_0=4$ , select the expression for the grid points $x_k$ , for $k=1, 2,..., n$ ?

$4+(k-1)\Delta x$ $4+k\Delta x$ $k$ $k\Delta x$ None of the above

(iv) What is $x_k^*$ for $k=1,2,..,n$ .

$\frac {x_{k-1}+x_k}{2}$ $x_k$ $x_{k-1}$ $x_0$ None of the above

(v) Evaluate this Riemann sum

$f(5)+f(6)+f(7)+f(8$ $f(6)+f(7)+f(8))$ $f(4)+f(5)+f(6)+f(7)$ None of the above

(vi) This Riemann sum is

midpoint Riemann sun right Riemann sum left Riemann sum None of the above

Determine the following indefinite integrals. Check your work by differentiation.

(a) $\int \frac {4+x^2}{1+x^2}dx=\answer {3\arctan (x)+x }+C$

(b) $\int \frac {\cos ^3(\theta )+1}{\cos ^2(\theta )}d\theta =\answer {\sin (\theta )+\tan (\theta )}+C$

(d) $\int 5x(x^2-3x)dx=\answer {\frac {5}{4}x^4-5x^3}+C$

(e) $\int \frac {x^2-3\sqrt {x}}{\sqrt [3]{x}}dx = \answer {\frac {3}{8}x^{\frac {8}{3}}-\frac {18}{7}x^\frac {7}{6}}+C$

(f) $\int (\sec ^2(\theta )+\sec (\theta )\tan (\theta ))d\theta =\answer {\tan (\theta )+\sec (\theta )}+C$

The figure shows the graph of a function $f$ . At what point(s) $c$ does the conclusion of the Mean Value Theorem hold for $f(x)$ on the interval $[4,8]$ ? (c is an integer)

$4$ $5$ $6$ $7$

Consider the following limit of Riemann sums for a function $g$ on $[0, \pi ]$ .

$\lim _{n\to \infty }\sum _{k=1}^n x_k^* \cos (x_k^*)\Delta x_k$

Express the limit as a definite integral.

$\int _0^n(-\frac {\sin (x^2)}{2})dx$ $\int _0^\pi (-\frac {\sin (x^2)}{2})dx$ $\int _0^\pi x\cos (x)dx$ $\int _0^n x\cos (x)dx$ None of the above

Consider an object moving along a straight line with the velocity $v(t)=12-3t$ on $[0,6]$ . Find the distance traveled over the given interval.

$\int _0^6(12-3t)dt$ $\int _0^6(3t-12)dt$ $\int _0^4(12-3t)dt+\int _4^6(3t-12)dt$ $\int _0^4(12-3t)dt-\int _4^6(3t-12)dt$ None of the above

Suppose that $f(x)\geq 0$ on $[1,3]$ , $f(x)\leq 0$ on $[3,5]$ , $\int _1^3f(x)dx=4$ , and $\int _3^5f(x)dx=-9$ .

(i) Evaluate the following integrals

(a) $\int _1^5 3f(x)dx=\answer {-15}$

(b) $\int _1^5 |f(x)|dx=\answer {13}$

(ii) Assume that $f$ is odd. Evaluate $\int _{-5}^{-3}f(x)dx=\answer {9}$

(ii) Assume that $f$ is even. Evaluate $\int _{-5}^{-3}f(x)dx=\answer {-9}$

Given that $\int _0^3f(x)dx=4$ , and $\int _3^6f(x)dx=-6$ , evaluate the following integrals:

(a) $\int _3^0 5f(x)dx=\answer {-20}$

(b) $\int _0^6 f(x)dx=\answer {-2}$

The acceleration function (in m/ $s^2$ ) is for a particle moving along a line is given by $a(t)=2t-\sin (t)$ , $0\leq t\leq 8$ , $v(0)=3, s(0)=4$ .

(a) Find the velocity at time t: $v(t)=\answer {t^2+\cos (t)+2} \frac {m}{s}$

(b) Find the distance traveled during the given time interval: $\answer {\frac {560}{3}+\sin (8)} m$

The acceleration function (in m/ $s^2$ ) is for a particle moving along a line is given by $a(t)=2t-4$ , $0\leq t$ , $v(0)=3, s(0)=0$ .

(a) Find the velocity at time t: $v(t)=\answer {t^2-4t+3} \frac {m}{s}$

(b) Find the position at time t: $s(t)=\answer {\frac {1}{3}t^3-2t^2+3t}$

Find the particular solution of $f'(x)=3x^2-1$ which satisfies the intial condition $f(2)=1$ . $f(x)=\answer {x^3-x-5}$

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

(a) Use geometry to evlaluate $\int _1^9 f(x)dx=\answer {32}$

(b) Select the graph of a rectangle whose net area is equal to $\int _1^9 f(x)dx=32$

(ii) The graph of $f$ on the interval $[0,6]$ is given in the figure

(a) Use geometry to evaluate $\int _0^6 f(t)dt=\answer {6-\pi }$

(b) Compute $\int _0^6 |f(t)|dt=\answer {\pi +6}$

An object is moving along a straight line. The graphs of its position function $s$ , its velocity $v$ , and its acceleration $a$ , on the time interval $[1,3]$ are given below.

Which curve is which: $s=\answer [format=string]{c}$ ; $v=\answer [format=string]{a}$ ; $a=\answer [format=string]{b}$

Consider an object moving along a line. The graph of the velocity, $v$ , of the object is shown in the figure below.

Select the graph of $s(t)$ given that $s(0)=0$