Honors Calculus AB Name:____Answer Key_______ Ch 4 Review of Applications of the Derivative 7 December 2010

Chapter 4 of the text focused on practical uses of derivatives. The topics included:

!

" f (x) to determine where the function is increasing/decreasing and possible extrema

!

" " f (x) to determine where the function is concave up/down and possible inflection points continuous function f has an tangent slope equal to average slope (Mean Value Theorem) function optimization by using situational constraints and finding extrema (Optimization) function approximations using tangent lines (Linear Approximation) limit evaluation of ratios using limits of the numerator and denominator (L’Hôpital’s Rule) derivatives to relate rates of change for different factors within a situation (Related Rates)

Here are some problems to do as a review of the ideas: 1. If a rectangle has it base on the x-axis and two vertices on the curve

!

y = e"x

2

, show that the rectangle has the largest possible area when the two vertices are at the points of inflection of the curve. Max area and inflection points at

!

x = ± 1

2 because A = y’.

2. a. Let f(x) = ax2 + bx + c, a ≠ 0, be a quadratic polynomial. How many points of inflection

does the graph of f have? Use calculus to justify your answer. f'’(x) = 0 only when a = 0, not quadratic if a = 0 so no points of inflection.

b. Let f(x) = ax3 + bx2 + cx + d, a ≠ 0, be a cubic polynomial. How many points of inflection

does the graph of f have? Use calculus to justify your answer. f’’(x) = when x = –b/3a so only one possible point of inflection.

c. Suppose the function y = f(x) satisfies the equation

!

dy

dx= ky 1"

y

L

#

$ %

&

' ( where k and L are

positive constants. Show that the graph of f has a point of inflection at the point where y = L/2. (This equation is called the logistic differential equation.) Using implicit differentiation, can determine that the second derivative is zero when y = 0, L, and L/2 but the first two are actually zeros of the first derivative and boundaries of y.

3. Consider a room in the shape of a cube, 4 meters on each side. A bug at

point P wants to walk to point Q at the opposite corner, as shown in the figure. Use calculus to determine the shortest path. Can you solve the problem without calculus? The shortest path is

!

4 5 when spider hits the corner halfway down the wall.

4. Find the point on the graph of

!

y =1

1+ x2

where the tangent line has the greatest slope, and the point where the tangent line as the least slope. The extrema occur at

!

x = ± 1

3

P

Q

4 m 4 m

4 m

2

1

-1

- 4 - 2 2 4

Ch. 4 Review and Extension Problems page 2

5. Find the point on the parabola y = 1 – x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area. The point is

!

1

3, 23( )

6. Find the highest and lowest point on the curve x2 + xy + y2 = 12. The points are (–2, 4) and (2, –4). 7. The regular square pyramid shown has all edges equal. The base is a square and the volume

of the pyramid is given by:

V =2

6L3.

Suppose all the edges are growing at a constant rate of 2 inches per second. At the particular moment that the base area is growing at a rate of 10 square inches per second how fast is the volume changing?

Use dB/dt to find L and then use L to find dV/dt =

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25 2

4

8. State a closed interval over which the function

!

f ( x) = x

1

3 does not satisfy the Mean Value Theorem and explain why. f(x) is not differentiable at x = 0 so any interval containing x = 0 will not work with the MVT.

2.5

2

1.5

1

0.5

1 2

2.5

2

1.5

1

0.5

1 2

2.5

2

1.5

1

0.5

1 2

L L

LL

Ch. 4 Review and Extension Problems page 3

9. Sketch a possible graph of a continuous function f that has the given properties: a) Domain of [0, 6] b) f(0) = 2 c) Domain of

!

" f is

!

[0,4)" (4,6] and it is pictured at right. Function needs closed endpoints at (0, 2) and (6, ?). It should have a local max at x = 1, local mins at x = 3 and 5.5, a local max that is a corner at x = 4 and a point of inflection at x = 2.

10. A continuous function f defined for all real numbers has the following properties:

•

!

f (0) =1,

!

f (±1) = 0 ,

!

f (2) =1. • lim

x!"f (x) = 2 and lim

x!"#f (x) = "1 .

•

!

f//(x) > 0on the intervals (!", 0),(0,2) .

•

!

f//(x) < 0on the interval (2,!) .

•

!

f/(x) > 0on the intervals (!", 0),(1,") .

•

!

f/(x) < 0on the interval (0,1).

Sketch a graph of f, labeling the relevant

points. Note: Corner at x = 0. 11. Consider the cubic polynomial

!

p(x) = ax3

+ bx2

+ c , where a 0! . Use calculus to analyze the behavior of the graph of

!

p(x) with differing values of “b”. Fill in the table below with your results. Use the second derivative test to determine if each of the extrema are a max or min.

Value of the “b” coefficient.

Location of Inflection Point(s)

Number and Location of Relative Extrema.

Relative Extrema – Max or Min

b = 0

x = 0

None

b < 0

x = –b/3a

x = 0 and x = –2b/3a x = 0 max

x = –2b/3a min

b > 0

x = –b/3a

x = 0 and x = –2b/3a x = 0 min

x = –2b/3a max

Ch. 4 Review and Extension Problems page 4

P

!

3

5

θ

S

12. The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function f. In the figure

at right,

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f (t) = "1

4t3

+3

2t2

+1 for

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0 " t " 4 and f is piecewise linear for

!

4 " t " 24 .

a. Find

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" f (22). Indicate units of measure. –3 Cal/min2

b. For the time interval

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0 " t " 24 , at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer.

Max f’ = 3 Cal/min when f’’= 0 at t = 2. The next two problems involve angles so a few angle sum identities are given in case they are helpful.

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sin(" ± #) = sin(")cos(#) ± cos(")sin(#)

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cos(" ± #) = cos(")cos(#) m sin(")sin(#)

!

tan(" ± #) =tan(") ± tan(#)

1m tan(")tan(#)

13. Find the maximum area of a rectangle that can be circumscribed about a 3x5

rectangular notecard. (The θ and the references to trig are hints.) Max at θ = π/4 where l = w =

!

4 2 14. An observer stands at a point P, one unit away from a straight track. Two runners start at

point S in the figure and run along the track. One runner runs three times as fast as the other. a. Find the maximum value of the observer’s angle of sight θ between the runners. (Hint:

Consider two triangles, PSA and PSB. Triangle PSA has legs 1 and a, and observer angle α. Triangle PSB has legs 1 and b, and observer angle β. How are angles α and β related to θ = β – α? How are legs a and b related to each other (b = 3a)?) Max θ = π/6

b. When runner A is 4 units from the stationary observer and running along the track at a

speed of 2 units per minute, how fast is runner B moving away from the observer?

d(PB)/dt =

!

9 15

34units per minute

Minutes

t

4 8

3

6

9

12

12

15

2016 24

(4, 9) (12, 9)

(16, 15) (20, 15)

(24, 3)

A B

1