Homework Homework Assignment #4 Read Section 5.5 Page 335, Exercises: 1 – 49(EOO) Rogawski...

Homework

Homework Assignment #4 Read Section 5.5 Page 335, Exercises: 1 – 49(EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 3351. Write the area function of f (x) = 2x + 4 with lower limit a = –2 as an integral and find a formula for it.


222

2

2 4 2 4 2 4 2

4 4

x x dx F x F x x

x x

Homework, Page 335


15. Find 1 , 0 , and , where tan4xG G G G x tdt

1

1 11 tan 0 tan tan

0 tan 0 0, tan 14 4

1 0, 0 0, 14

xdG tdt G x tdt x

dx

G G

G G G

Homework, Page 335Find formulas for the functions represented by the integrals.


2

19. x tdt

2

2 22 42

1

1 11

2 2 2x

x xtdt F x F

Homework, Page 335Find formulas for the functions represented by the integrals.


2

413. secx d

2

4

2

4

sec 4

tan tan tan 14

sec tan 1

x

x

d F x F

x x

d x

Homework, Page 335Express the antiderivative F(x) of f (x) satisfying the given initial condition as an integral.


17. sec , 0 0x F

0 secxF x tdt

Homework, Page 335Calculate the derivative.


10021. cos5tdxdx

dt

100 cos5 cos5tdxdx t

dt

Homework, Page 33525. Make a rough sketch of the graph of the area function of g(x) shown in Figure 12.


1 2 3 4



2 2029. sinxd

tdtdx

2

2

2 2 2 2 20

2 2 20

sin sin 2 2 sin

sin 2 sin

x

x

dtdt x x x x

dx

dtdt x x

dx



30 233. sinx

dtdt

dx

3

3

3

0 2 2 2 3 2 2 2 30

0 2 2 2 3

sin sin sin 3 3 sin

sin 3 sin

x

x

x

d dtdt tdt x x x x

dx dx

dtdt x x

dx

Homework, Page 335

37. Find the min and max of A(x) on [0, 6].


0 2Let and with as in Figure 13.x xA x f t dt B x f t dt f x

min 1.25,max 1.25

Homework, Page 33541. Area Functions and Concavity. Explain why the following statements are true. Assume f (x) is differentiable.

(a) If c is an inflection point of A(x), then f ′(c) = 0.

Since f(x) = A′(x), f ′(x) =A″(x) and A″(x) = 0 at inflections points of A(x).

(b) A(x) is concave up if f (x) is increasing.

If f(x) is increasing, then f ′(x) > 0, hence A″(x) > 0 and A(x) is concave up.

(c) A(x) is concave down if f (x) is decreasing.

If f(x) is decreasing, then f ′(x) < 0, hence A″(x) < 0 and A(x) is concave up.


Homework, Page 33545. Sketch the graph of an increasing function f (x) such that both f ′(x) and A (x) are decreasing.


Homework, Page 33549. Determine the function g (x) and all values of c such that


2 6x

c g t dt x x

2

2

6 2 1

6 0 2 3 0 3,2

2 1, 3,2

dx x x g x

dx

x x x x c

g x x c


Chapter 5: The IntegralSection 5.5: Net or Total Change as the

Integral of a Rate

Jon Rogawski

Calculus, ETFirst Edition


If water enters an empty bucket at the rate r(t), as shown in Figure 1, then the amount of water in the bucket at any time on the interval [0, 4]is equal to the area under the curve up to the vertical line at the timein question.

0

Since the water enters the bucket at a varying rate, we can use

the FTC to find the amount of water in the bucket at any time ,

specifically: 0 . Since the bucket is initially

empty, the amo

x

x

r t dt R x R unt of water in the bucket is at time is ( ) and the

integral gives us the net change in the amount of water in the bucket.

x R x


If s′(t) is both positive and negative on [t1, t2], the integral will total the signed areas and give us the net change in s. This is the basis of Theorem 1.

Example, Page 3414. A survey shows that a mayoral candidate is gaining votes at the rate of 2000t + 1000 votes per day since she announced her candidacy. How many supporters will the candidate have after 60 days, assuming she had no supporters at t = 0.



The table shows the flow rate of cars passing an observation point inunits of cars per hour. Estimate the number of cars using the highway in the two hour period by taking the average of the left and right Riemann sums.


Remembering that displacement is a vector quantity, the integral of velocity, another vector quantity, over an interval of time gives us the net displacement. To find distance traveled, we must take the integralof the absolute value of velocity, as illustrated in Figure 2.

Figure 3 illustratesthe path of theparticle with velocityv(t) from Figure 2.


Theorem 2 formalizes what we observed in Figure 2.

Example, Page 3418. A projectile is released with initial vertical velocity 100 m/s. Use the formula v(t) = 100 – 9.8t for velocity to determine distance traveled in the first 15 seconds.


Example, Page 34118. Suppose that the marginal cost of producing x video recorders is 0.001x2 – 0.6x + 350 dollars. What is the cost of producing 300 units? If production is set at 300 units, what is the cost of producing 20 additional units?


Homework

Homework Assignment #5 Read Section 5.6 Page 341, Exercises: 1 – 19(Odd)


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Homework Homework Assignment #4 Read Section 5.5 Page 335, Exercises: 1 – 49(EOO) Rogawski...

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