5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 317

TIP • If f′′(a)=0,f may or may not have a point of inflection atx=a, e.g., as in the
functionf(x)=x^4 ,f′′( 0 )= 0 butatx=0,fhas an absolute minimum.

Example 2

If the marginal cost of producingxunits of a commodity isC′(x)= 5 + 0. 4 x,

find (a) the marginal cost whenx=50;

(b) the cost of producing the first 100 units.

Solution:

(a) Marginal cost atx=50:

C′(50)= 5 + 0 .4(50)= 5 + 20 = 25.

(b) Cost of producing 100 units:

C(t)=

∫ 1

0

C′(x)dx

=

∫ 100

0

(5+ 0. 4 x)dx

= 5 x+ 0. 2 x^2

] 100

0

=

(

5(100)+ 0 .2(100)^2

)

− 0 = 2500.

Temperature Problem

Example

On a certain day, the changes in the temperature in a greenhouse beginning at 12 noon are

represented byf(t)=sin

(

t

2

)

degrees Fahrenheit, wheretis the number of hours elapsed

after 12 noon. If at 12 noon, the temperature is 95◦F, find the temperature in the greenhouse

at 5 p.m.

LetF(t) represent the temperature of the greenhouse.

F( 0 )= 95 ◦F

F(t)= 95 +

∫ 5

0

f(x)dx

F( 5 )= 95 +

∫ 5

0

sin

(

x

2

)

dx

= 95 +

[

−2 cos

(

x

2

)] 5

0

= 95 +

[

−2 cos

(

5

2

)

−(−2 cos( 0 ))

]

= 95 + 3. 602 = 98. 602

The temperature in the greenhouse at 5 p.m. is 98.602◦F.