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

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

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

Solution:

(a) The particle is moving to the right whenv(t)>0.

Entery 1 =3 cos(2x).Obtainy 1 =0 whent=

π

4

,

3 π

4

,

5 π

4

, and

7 π

4

.

The particle is moving to the right when:

0 <t<

π

4

,

3 π

4

<t<

5 π

4

,

7 π

4

<t< 2 π.

(b) The particle stops whenv(t)=0.

Thus, the particle stops att=

π

4

,

3 π

4

,

5 π

4

, and

7 π

4

.

(c) Total distance traveled

∫ 2 π

0

∣∣

3 cos(2t)

∣∣

dt.

Enter

∫

(abs(3 cos(2x)),x,0,2π)and obtain 12.

Thus, the total distance traveled by the particle is 12.

14.3 Definite Integral as Accumulated Change

Main Concepts:Business Problems, Temperature Problem, Leakage Problem,

Growth Problem

Business Problems

Profit=Revenue−Cost P(x)=R(x)−C(x)

Revenue=(price)(items sold) R(x)=P(x)

Marginal Profit P′(x)

Marginal Revenue R′(x)

Marginal Cost C′(x)

P′(x),R′(x), andC′(x) are the instantaneous rates of change of profit, revenue, and cost,

respectively.

Example 1

The marginal profit of manufacturing and selling a certain drug isP′(x)= 100 − 0. 005 x.

How much profit should the company expect if it sells 10,000 units of this drug?

P(t)=

∫ 1

0

P′(x)dx

=

∫10, 000

0

(100− 0. 005 x)dx= 100 x−

0. 005 x^2

2

]10, 000

0

=

(

100(10, 000)−

0. 005

2

(10, 000)^2

)

=750, 000