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

MA 3972-MA-Book April 11, 2018 15:57

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

Set 2k^2 +k− 6 = 30 ⇒ 2 k^2 +k− 36 = 0

⇒(2k+9)(k−4)=0ork=−

9

2

ork= 4.

Sincek>0,k= 4.

Example 4

Iff′(x)=g(x), andgis a continuous function for all real values ofx, express

∫ 5

2

g(3x)dx

in terms off.

Letu= 3 x;du= 3 dxor

du

3

=dx.

∫

g(3x)dx=

∫

g(u)

du

3

=

1

3

∫

g(u)du=

1

3

f(u)+C

=

1

3

f(3x)+C

∫ 5

2

g(3x)dx=

1

3

f(3x)

] 5

2

=

1

3

f(3(5))−

1

3

f(3(2))

=

1

3

f(15)−

1

3

f(6)

Example 5

Evaluate

∫ 4

0

1

x− 1

dx.

Note that you cannot evaluate using the First Fundamental Theorem of Calculus since

f(x)=

1

x− 1

is discontinuous atx=1.

Example 6

Using a graphing calculator, evaluate

∫ 2

− 2

√

4 −x^2 dx.

Using a TI-89 graphing calculator, enter

∫(√

( 4 −x∧ 2 ), x, −2, 2

)

and obtain 2π.

Second Fundamental Theorem of Calculus

Iff is continuous on [a,b] andF(x)=

∫x

a

f(t)dt, thenF′(x)=f(x) at every pointxin

[a,b].