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

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

340 STEP 4. Review the Knowledge You Need to Score High

Step 9: Verify result by differentiating:

dy

dx

=

x^2

2

− 5 x− 2

d^2 y

dx^2

=x− 5.

Part B Calculators are allowed.

13. Average value=

1

π/ 3 −π/ 4

∫π/ 3

π/ 4

tanxdx.

Enter=(1/(π/ 3 −π/4))

∫

(tanx,x,π/4,π/3)

and obtain

6 ln(2)

π

= 1. 32381.

14. v(t)=

∫

a(t)dt

=

∫

3 e^2 t=

3

2

e^2 t+C

v(0)=

3

2

e^0 +C=

1

2

⇒

3

2

+C=

1

2

orC=− 1

v(t)=

3

2

e^2 t− 1

Displacement=

∫ 3

0

(

3

2

e^2 t− 1

)

dt.

Enter

∫

( 3 / 2 ∗e∧(2x)−1,x,0,3)

and obtain 298. 822.

Distance traveled=

∫ 3

0

∣∣

v(t)

∣∣

dt.

Since

3

2

e^2 t− 1 >0 fort≥0,

∫ 3

0

∣∣

v(t)

∣∣

dt=

∫ 3

0

(

3

2

e^2 t− 1

)

dt= 298. 822.

15. Step 1: y(t)=y 0 ekt

y(1)= 5000 ⇒ 5000 =y 0 ek⇒y 0

= 5000 e−k

y(3)= 4000 ⇒ 4000 =y 0 e^3 k

Substituting:

y(0)= 5000 e−k, 4000=(5000e−k)e^3 k

4000 = 5000 e^2 k

4

5

=e^2 k

ln

(

4

5

)

=ln

(

e^2 k

)

= 2 k

k=

1

2

ln

(

4

5

)

≈− 0. 111572.

Step 2: 5000=y 0 e−^0.^111572

y(0) =(5000)/e−^0.^111572 ≈ 5590. 17

y(t)=( 5590. 17 )e−^0.^111572

Step 3: y(7)=(5590.17)e−^0 .111572(7)≈ 2560

Thus, sales for the 7th month are

approximately 2560 units.

16. Step 1: Separate the variables:
    dy
    dx

=

2 y

x+ 1

dy

2 y

=

dy

x+ 1

.

Step 2: Integrate both sides:

∫

dy

2 y

=

∫

dx

x+ 1

1

2

ln|y|=ln

∣∣

x+ 1

∣∣

+C.

Step 3: Substitute given value (0, 4):

1

2

ln(4)=ln(1)+C

ln 2=C

1

2

ln|y|−ln|x+ 1 |=ln 2

ln

∣∣

∣∣y

1 / 2

x+ 1

∣∣

∣∣=ln 2