5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:5

More Applications of Definite Integrals 335

4. How much money should Mary invest at 7.5% interest a year compounded
   continuously, so that she will have $100,000 after 20 years.
   Answer: y(t)=y 0 ekt,k= 0 .075, andt= 20 .y(20)=100, 000=y 0 e(0.075)(20). Thus,
   using a calculator, you obtainy 0 ≈22313, or $22,313.


5. Given
   dy
   dx

=

x

y

andy(1)=0, solve the differential equation.

Answer: y d y=xdx⇒

∫

ydy=

∫

xdx⇒

y^2

2

=

x^2

2

+c⇒ 0 =

1

2

+c⇒c=−

1

2

Thus,

y^2

2

=

x^2

2

−

1

2

ory^2 =x^2 −1.

6. Identify the differential equation for the
   slope field shown.
   Answer: The slope field suggests a hyperbola
   of the formy^2 −x^2 =k,so2y
   dy
   dx
   − 2 x= 0

and

dy

dx

=

x

y

.

14.7 Practice Problems

Part A The use of a calculator is not

allowed.

1. Find the value ofcas stated in the Mean
   Value Theorem for Integrals forf(x)=x^3
   on [2, 4].


2. The graph of fis shown in Figure 14.7-1.
   Find the average value off on [0, 8].

(4, 4)

1

012345678

2

3

4

x

y

f

Figure 14.7-1