Deck 4: Using the Derivative

Question 1

Consider a continuous function with the following properties: $f(0)=3$ $\left|f^{\prime}(x)\right|<0.5$ $f^{\prime \prime}(x)<0$ for $x<0$ $f^{\prime}(1)=0$ . Which of the following are inconsistent with these four conditions?

A) $\lim _{x \rightarrow-\infty} f(x)=0$
B) $f(1)=$ 5
C) $f^{\prime \prime}(1)=0$

Answer 1

$\lim _{x \rightarrow-\infty} f(x)=0$
$f(1)=$ 5

Question 2

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Suppose that you are told that $f(0)$ = 5.Estimate f(2).

A)3
B)3.5
C)5
D)6.5

Answer 2

3.5

Question 3

Starting at time t = 0, water is poured at a constant rate into an empty vase (pictured below).It takes ten seconds for the vase to be filled completely to the top.Let h = f(t)be the depth of the water in the vase at time t.For what value of h is the largest?

Answer 3

6

Question 4

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Where in the interval 0 $\le$ x $\le$ 4 does f achieve its global maximum?

A)0
B)1
C)2
D)3
E)4

Question 5

Given below are the graphs of two functions f(x)and g(x).Let .Is increasing or decreasing on the interval 3 < x < 4?

Question 6

For which interval(s)is the function $f(x)=x^{5}-$ 16x³ decreasing?

A) $x<$ -4 or 4 $<$$x$
B)-4 $<$$\chi$$<$ 4
C)-4 $<$$x$$<$ 0
D)0 $<$$x$$<$ 4
E)-4 $<$$x$$<$ 4,
$x \neq 0$

Question 7

Consider a continuous function with the following properties: $f(0)=4$ $\left|f^{\prime}(x)\right|<0.5$ $f^{\prime \prime}(x)<0$ for $x<0$ $f^{\prime}(1)=0$ . Which of the following is true?

A)The graph must have a local maximum for $x<0$ .
B)The graph must not have a local maximum for $x<0$ .
C)The graph may or may not have a local maximum for $x<0$ .

Question 8

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Which of the following values of x are local minima of f?

A)2
B)0
C)4
D)3
E)1

Question 9

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Is f increasing or decreasing on the interval $2 \leq x \leq 3$ ?

Question 10

A water tank is constructed in the shape of a sphere seated atop a circular cylinder.If water is being pumped into the tank at a constant rate, let $h(t)$ be the height of the water as a function of time.Which of the following is true at the point where $h(t)=$ b?

A)The second derivative doesn't exist
B)There is an inflection point
C)The slope is infinite
D)None of the above

Question 11

Sketch a graph of a function whose at x=-1, < 0 when x< -1, < 0 when x> -1,

Question 12

Sketch a graph of a function whose at x=1, < 0 when x< 1, < 0 when x> 1, Is it possible to have at any value from -2

Question 13

A particle is travelling along the x-axis according to the function ( t-3 )( t-1 )^2.When is the velocity of the particle equal to 0?

Question 14

A particle is travelling along the x-axis according to the function ( t-3 )( t-2 )².Assuming t 0, when is the acceleration of the particle equal to 0?

Question 15

A water tank is constructed in the shape of a sphere seated atop a circular cylinder.If water is being pumped into the tank at a constant rate, let $h(t)$ be the height of the water as a function of time.For the interval a < t < b, which of the following is true?

A) $h(t)$ is concave down
B) $h(t)$ is linear
C) $h(t)$ is concave up

Question 16

Starting at time t = 0, water is poured at a constant rate into an empty vase (pictured below).It takes ten seconds for the vase to be filled completely to the top.Let h = f(t)be the depth of the water in the vase at time t.Is h = f(t)concave up or down on the region 3 < t < 9?

Question 17

Given below are the graphs of two functions f(x)and g(x).Let .Is the point x = 1 a local minimum, a local maximum, or neither for the function ?

Question 18

Given below are the graphs of two functions f(x)and g(x).Graph on a similar set of axes.

Question 19

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Suppose that you are told that $f(0)$ = 3.Which of the following is an exact expression for $f(2)$ ?

A) $\int_{3}^{2} f^{\prime}(x) d x$
B) $3-\int_{0}^{-2} f^{\prime}(x) d x$
C) $\int_{0}^{2} f^{\prime}(x) d x+3$
D) $\int_{0}^{2} f^{\prime}(x) d x-3$

Question 20

Determine the equation of the tangent line at x=0 and the value of f(x)at x=1.5 given $f(x)=a^{x}$ for all values of a

A)y=(ln a)x+1, y=1.5(ln a)+1
B)y=(ln a)(a^1.5)x-1, y=-1
C)y=(a^1.5)[(ln a)x-1], y=-1
D)y=(ln a)x-1, y=1.5(ln a)-1
E)y=(ln a)x-1, y=1.5

Question 21

Consider the function , for .Is f increasing or decreasing at x = 3.82?

Question 22

Let f be a function.Is it true or false that the inflection points of f are the local extrema of f '.

Question 23

Given the table of data about the second derivative of a function f, which of the following types of a function could f be? Assume b > 0.The other constants can be positive or negative.
$\begin{array}{lcccc}x & 0 & 1 & 2 & 3 \\& 1 & -2 & -5 & -8 \\f^{\prime \prime}(x) & & & &\end{array}$

A) $a e^{b x}$
B) $e^{-x^{2} / b}$
C)a quadratic (i.e. $a x^{2}+b x+c$ )
D)a cubic (i.e. $c x^{3}+b x^{2}+c x+d$ )
E) $\sin (b x)$

Question 24

Consider the function , for .What is the largest value of f? Round to 2 decimal places.

Question 25

Consider the function for .What is largest value of a such that on the region ?

Question 26

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Which of the following values are inflection points of f?

A)0
B) $\frac{\pi}{6}$
C) $\frac{\pi}{2}$
D) $\frac{5 \pi}{6}$
E) $\frac{3 \pi}{2}$

Question 27

Consider $f(x)=x^{2} e^{-x}$ for $-1 \leq x \leq 3$ .For which value(s)of x is $f(x)$ least?

A)0.586
B)-1
C)0
D)3

Question 28

Let $f(x)=e^{\frac{-x^{2}}{b}}$ , where b is a positive constant.Which of the following are inflection points of f?

A)b
B) $-b$
C) $\sqrt{\frac{b}{2}}$
D) $-\sqrt{\frac{b}{2}}$
E)0

Question 29

Consider for .Is f increasing or decreasing on the interval 0 < x < 2?

Question 30

Let f(x)be a function with positive values and let $g=\sqrt{f}$ .If f has a local maximum at $x=x_{0}$ , what about g?

A)g has a local maximum at $x=x_{0}$
B)g has a local minimum at $x=x_{0}$
C)g could have a local maximum or a local minimum at $x=x_{0}$

Question 31

Consider the function $f(x)=\frac{1}{1+a e^{-x}}$ , for $a>0$ .As a increases, what happens to the horizontal asmyptotes of f?

A)They shift upward.
B)They remain the same.
C)They shift downward.

Question 32

Below is the graph of the rate r at which people arrive for lunch at Cafeteria Charlotte.Checkers start at 12:00 noon and can pass people through at a constant rate of 5 people/minute.Let f(t)be the length of the line (i.e.the number of people)at time t.Suppose that at 11:50 there are already 150 people lined up.Using the graph together with this information, sketch a graph of f.

Question 33

Below is the graph of the rate r at which people arrive for lunch at Cafeteria Charlotte.Checkers start at 12:00 noon and can pass people through at a constant rate of 5 people/minute.Let f(t)be the length of the line (i.e.the number of people)at time t.Suppose that at 11:50 there are already 150 people lined up.Using the graph together with this information, is the time 12:34 a local minimum, a local maximum, or neither of f?

Question 34

Consider the function , for .Graph the function and use your graph to find how many roots there are to the equation .

Question 35

Below is the graph of the rate r at which people arrive for lunch at Cafeteria Charlotte.Checkers start at 12:00 noon and can pass people through at a constant rate of 5 people/minute.Let f(t)be the length of the line (i.e.the number of people)at time t.Suppose that at 11:50 there are already 150 people lined up.Using the graph together with this information, is f concave up or down on the interval 11:55 < t < 12:07?

Question 36

Consider the two-parameter family of curves $y=\alpha x+\frac{b}{x}$ , with $a>0$ and $b>0$ .Is the graph concave up or down at the point x = -2?

A)Concave down
B)Concave up
C)It depends on the values of a and b

Question 37

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Where is f increasing most rapidly?

A)0
B) $\frac{\pi}{6}$
C) $\frac{3 \pi}{2}$
D) $\frac{5 \pi}{6}$
E) $\frac{\pi}{2}$

Question 38

Below is the graph of the rate r at which people arrive for lunch at Cafeteria Charlotte.Checkers start at 12:00 noon and can pass people through at a constant rate of 5 people/minute.Let f(t)be the length of the line (i.e.the number of people)at time t.Suppose that at 11:50 there are already 150 people lined up.Using the graph together with this information, when is the line the longest?

Question 39

Consider the function for .For what value of x on the region does f have a local maximum? If there is more than one value, give the smallest one.

Question 40

Graph the function for .

Question 41

What does the Extreme Value Theorem allow us to conclude about f if f is continuous on [-10, 80]? Mark all that apply.

A)f has a global maximum on [-10, 80].
B)f has a global minimum on [-10, 80].
C)f has an inflection point on [-10, 80].
D)None of the above

Question 42

The revenue for selling q items is $R(q)=600 q-2 q^{2}$ and the total cost is C(q)= 110 + 60q.Which function gives the total profit earned?

A) $-540 q+2 q^{2}+110$
B) $-540 q+2 q^{2}-110$
C) $540 q-2 q^{2}+110$
D) $540 q-2 q^{2}-110$

Question 43

A single cell of a bee's honey comb has the shape shown.The surface area of this cell is given by $A=6 h s+\frac{3}{2} s^{2}\left(\frac{-\cos \theta}{\sin \theta}+\frac{\sqrt{3}}{\sin \theta}\right)$ where h, s, $\theta$ are as shown in the picture.Keeping h and s fixed, for what angle, $\theta$ , is the surface area minimal? Round to the nearest one tenth of a degree.

Question 44

Write a formula for total cost as a function of quantity r when fixed costs are $30,000 and variable costs are $1,600 per item.

Question 45

When light strikes a shiny surface, much of it is reflected in the direction shown.However some of it may be scattered on either side of the reflected light.If the intensity (brightness)of the scattered light at the angle $\theta$ (shown in the picture)is I, the Phong model says that $I=k \cos ^{n}(\theta)$ where k and n are positive constants depending on the surface.Thus this function gives an idea of how "spread-out" the scattered light is.What effect does decreasing the parameter n have of the graph of I?

A)The graph drops less quickly
B)The graph stretches upward
C)The graph shrinks downward
D)The graph drops more quickly

Question 46

The cost C(q)(in dollars)of producing a quantity q of a certain product is shown in the graph below.Suppose that the manufacturer can sell the product for $2.50 each (regardless of how many are sold), so that the total revenue from selling a quantity q is R(q)= 2.5q.The difference is the total profit.Let be the quantity that will produce the maximum profit.What is ?

Question 47

The cost C(q)(in dollars)of producing a quantity q of a certain product is shown in the graph below.The average cost is given by $a(q)=\frac{C(q)}{q}$ .Graphically, a(q)is the slope of the line between which two points?

A)(0, 0)
B) $(q, C(q))$
C)(0, q)
D) $(0, C(q))$

Question 48

Suppose f is a cubic function with critical points at x = 8 and x = 6.What is the x-coordinate of the inflection point of f?

Question 49

A rectangle is inscribed between the function y= -x² + 49 and the x-axis.What is the maximum area of the square if the base if two vertices of the square lie on the x-axis?

A)3.5
B)257.25
C)428.75
D)8.042
E)40.459

Question 50

Total cost and revenue are approximated by the functions C = 1200 + 3.5q and R = 6q, both in dollars.Identify the marginal cost per item.

Question 51

Consider the one-parameter family of functions given by $e^{A x}+e^{-A x}$ for $A>0$ .What are the effects on the graph as the value of A is decreased?

A)The global minimum is decreased.
B)The global minimum is increased.
C)The curvature is decreased (i.e.the curve is wider)
D)The curvature is increased (i.e.the curve is narrower)

Question 52

The revenue for selling q items is and the total cost is C(q)= 120 + 60q.What quantity maximizes profit?

Question 53

If you want to maximize profit, you should minimize average cost.

Question 54

In the function y=2 sin (x)+1.96, in the interval from 0 $\leq x \leq$ $\pi$ , at which value(s)of x does the function contain a global maximum?

A) $\pi$
B)0
C) $\pi / 2$
D)4 $\pi$
E)2 $\pi$

Question 55

For and , what is the global maximum value of ?

Question 56

The cost C(q)(in dollars)of producing a quantity q of a certain product is shown in the graph below.The average cost is given by $a(q)=\frac{C(q)}{q}$ .Find on the graph the quantity $90$ where a(q)is minimal.Now suppose that the fixed costs (i.e., the costs of setting up before production starts)are doubled.Sketch the new cost function $C_{1}(g)$ on the same set of axes as the original one and let $q_{1}$ be the quantity where the new $a_{1}(g)$ is minimal.Which of the following is true?

A) $q_{0}$$<$$q_{1}$
B) $q_{0}>q_{1}$
C) $q_{0}=q_{1}$
D)Cannot be determined

Question 57

A window with a rectangular base is topped by a semicircle creating a Norman window.A plate of glass 24 ft² in area.What are the dimensions that would minimize the metal frame around the window? [round to 3 decimal places]
Base: _____________, Height: ________________, Radius: ___________________

Question 58

Total cost and revenue are approximated by the functions C = 1900 + 4q and R = 6q, both in dollars.Give a formula for the profit function.

Question 59

In the function y=-4 sin (x)+4.96, in the interval from 0 $\leq x \leq$ $\pi$ , what is the global maximum value?

A)0.960
B)8.96
C)4.960
D)0
E)1.571

Question 60

Sketch a graph of a function with two local minima, no global maximum, but a global minimum.

Question 61

Daily production levels in a plant can be modeled by the function , which gives units produced t hours after the factory opened at 8am.At what time during the day is factory productivity a maximum? Answer in the form "_:_ _" (without an "am" or "pm").

Question 62

If you throw a stone into the air at an angle of $\theta$ to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{100}\left(1+\tan ^{2} \theta\right)$ , where y is the height of the stone above the ground, x is the horizontal distance.If the angle $\theta$ is fixed, what value of x gives the maximum height? (Your answer will $\theta$ .)

A) $\frac{50 \tan \theta}{1+\tan ^{2} \theta}$
B) $\frac{1+\tan ^{2} \theta}{50 \tan \theta}$
C) $\frac{\tan \theta}{50+50 \tan ^{2} \theta}$
D) $\frac{50+50 \tan ^{2} \theta}{\tan \theta}$

Question 63

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm³ per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.

Question 64

The function gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

Question 65

A rectangular swimming pool is 10 meters long and 6 meters wide.It has a depth of 1 meter at the shallow end, then slopes to a depth of 1.5 meters at the deep end, as shown in the following cross section (not to scale).It is being filled with a hose at a rate of 50,000 cubic centimeters per minute.225 minutes after the hose is turned on, the water is rising at a rate of _____ cm per second.Round to 3 decimal places.

Question 66

The function gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

Question 67

Find the quantity q which maximizes profit if the total revenue, R(q), and the total cost, C(q), are given in dollars by , where units.
Round to the nearest whole number.

Question 68

If you throw a stone into the air at an angle of $\theta$ to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{160}\left(1+\tan ^{2} \theta\right)$ ,
where y is the height of the stone above the ground, x is the horizontal distance.Suppose the stone is to be thrown over a wall at a fixed horizontal distance l away from you.If you can vary $\theta$ , what is the highest wall that the stone can go over? (Your answer will contain l.)

Question 69

What is the shortest distance from the point (0,1)to the curve ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

Question 70

A fan is watching a 100-meter footrace from a seat in the bleachers 15 meters back from the midway point.The winning runner is moving approximately 8 meters per second.How fast is the distance from the fan to the winning runner changing when he is x meters into the race?

Question 71

A cupful of olive oil falls on the floor forming a circular puddle.Its radius is increasing at a constant rate of 0.2 cm/sec.What is the rate of increase in the area of the olive oil when its circumference measures 20 cm?

Question 72

A bar of ice cream, with dimensions of 3 cm by 3 cm by 3 cm placed on a mesh screen on top of a cylindrical funnel that is 6 cm high and 6 cm in diameter.If the ice cream is melting at a rate of 3.3 $\mathrm{cm}^{3} / \mathrm{min}$ into the funnel, what is the rate of change of the height of the funnel when half of the ice cream has melted? $V_{f u m a l}=\frac{1}{3} \pi r^{2} h$

A)0.232
B)0.032
C)0.132
D)0.284
E)0.083

Question 73

A submarine can travel 30mi/hr submerged and 60mi/hr on the surface.The submarine must stay submerged if within 200 miles of shore.Suppose that this submarine wants to meet a surface ship 200 miles off shore.The submarine leaves from a port 300 miles along the coast from the surface ship.What route of the type sketched below should the sub take to minimize its time to rendezvous? Give the value of y to 2 decimal places.

Question 74

A spherical lollipop has a circumference of 7.9 centimeters.A student decides to measure the rate of change of the volume of the lollipop, in per minute.The student licks the lollipop and measures the circumference every minute.The radius is decreasing at a rate of 0.18 cm/min.Determine the rate at which the volume is changing when the circumference is half of it's original size.[ ]

Question 75

A normal distribution in statistics is modeled by the function $N(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$ determine where the maximum value of the function would occur.

A)-2
B)-1
C)0
D)-2 $\pi$
E) $\ell$ ^-2

Question 76

Find the marginal cost for q = 100 when the fixed costs in dollars are 1000, the variable costs are $190 per item, and each sells for $310.

Question 77

The regular air fare between Boston and San Francisco is $600.An airline flying 747s with a capacity of 480 on this route observes that they fly with an average of 400 passengers.Market research tells the airlines' managers that each $20 fare reduction would attract, on average, 20 more passengers for each flight.How should they set the fare to maximize their revenue?

Question 78

The number of plants in a terrarium is given by the function , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Question 79

Air is being blown into a spherical balloon at a rate of 70 cm³ per second.At what rate is the surface area of the balloon increasing when the radius is 10 cm? Round to 2 decimal places, and do not include units.

Question 80

A Brian's candy sugar wand is made from flavored sugar inside a straw.The straw is 210 mm long and 5 mm in diameter.The child accidentally poked a hole in the bottom, making the height of the sugar fall at a rate of 1 mm per second.The child realizes that there is a hole after 1 seconds.What was the rate of change of the volume of the sugar at this time?
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