5 Steps to a 5 AP Calculus BC 2019

36 STEP 2. Determine Your Test Readiness

53. See Figure DS-14.

10–1 2 x

y y=x+ 2

y=x^2

Figure DS-14

To find points of intersection, setx^2 =x+ 2

⇒x^2 −x− 2 = 0 ⇒x=2orx=−1.

Area of cross section=((x+2)−x^2 )^2.

Volume of solid,V=

∫ 2

− 1

(

x+ 2 −x^2

) 2

dx.

Using your calculator, you obtain:V=

81

10

.

54. Separate and simplify
    dP
    dt

=. 35 P

(

1 −

P

4000

)

.

1

P

(

1 −

P

4000

)dP=. 35 dt

4000

P(4000−P)

dP=. 35 dt

Integrate with a partial fraction

decomposition.

∫

4000

P(4000−P)

dP=

∫

. 35 dt
∫
dP
P

+

∫

dP

4000 −P

=

∫

. 35 dt

ln|P|−ln

∣∣

4000 −P

∣∣

=. 35 t+C 1

ln

∣∣

∣

∣

P

4000 −P

∣∣

∣

∣=.^35 t+C^1

P

4000 −P

=C 2 e.^35 t

Population att=0 is 100, so

100

4000 − 100

=

100

3900

=

1

39

=C 2.

The population model is

P

4000 −P

=

e.^35 t

39

⇒ 39 P=e.^35 t(4000−P)

⇒ 39 P+e.^35 tP= 4000 e.^35 t

⇒P

(

39 +e.^35 t

)

= 4000 e.^35 t

⇒P=

4000 e.^35 t

39 +e.^35 t

⇒P=

4000

39 e−.^35 t+ 1

.

Whent=5,P=

4000

39 e−.35(5)+ 1

⇒P=

4000

39 e−^1.^75 + 1

≈ 514 .325.

55. If
    dy
    dx

=

−y

x^2

andy=3 andx=2, approximate

ywhenx=3. Use Euler’s

Method with an increment of 0.5.

y(2)=3 and

(

dy

dx

)

x=2,y= 3

=

− 3

4

so

y(2.5)=y(2)+ 0. 5

(

dy

dx

)

x=2,y= 3

= 3 + 0 .5(− 0 .75)= 2. 625

(

dy

dx

)

x= 2 .5,y= 2. 625

=

− 2. 625

(2.5)^2

=

− 21

8(6.25)

=

− 21

50

=− 0. 42

y(3)=y(2.5)+ 0. 5

(

dy

dx

)

x= 2 .5,y= 2. 625

= 2. 625 + 0 .5(− 0 .42)

= 2. 625 − 0. 21 = 2. 415.