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

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 273

(a) Evaluate:p(0),p(1),p(4).
(b) Evaluate:p(5),p(7),p(8).
(c) At what value oftdoesphave a maximum value?
(d) On what interval(s) ispdecreasing?
(e) Draw a sketch of the graph ofp.

Solution:

(a) p(0)=

∫ 0

0

f(t)dt= 0

p(1)=

∫ 1

0

f(t)dt=

(1)(4)

2

= 2

p(4)=

∫ 4

0

f(t)dt=

1

2

( 2 + 4 )( 4 )= 12

(Note:f(t) forms a trapezoid fromt=0tot=4.)

(b) p(5)=

∫ 5

0

f(t)dt=

∫ 4

0

f(t)dt+

∫ 5

4

f(t)dt

= 12 −

(1)(4)

2

= 10

p(7)=

∫ 7

0

f(t)dt=

∫ 4

0

f(t)dt+

∫ 5

4

f(t)dt+

∫ 7

5

f(t)dt

= 12 − 2 −(2)(4)= 2

p(8)=

∫ 8

0

f(t)dt=

∫ 4

0

f(t)dt+

∫ 8

4

f(t)dt

= 12 − 12 = 0

(c) Sincef≥0 on the interval [0, 4],pattains a maximum att=4.

(d) Sincef(t) is below thex-axis fromt=4tot=8, ifx>4,
∫x

0

f(t)dt=

∫ 4

0

f(t)dt+

∫ x

4

f(t)dtwhere

∫x

4

f(t)dt< 0.

Thus,pis decreasing on the interval (4, 8).

(e) p(x)=

∫x

0

f(t)dt. See Figure 13.1-5 for a sketch.

x 0 1 2 3 4 5 6 7 8

p(x) 0 2 6 10 12 10 6 2 0