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

MA 3972-MA-Book May 9, 2018 10:9

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

7.10 Cumulative Review Problems

(Calculator) indicates that calculators are
permitted.

26. Find lim
    h→ 0

sin

(

π

2

+h

)

−sin

(

π

2

)

h

.

27. Iff(x)=cos^2 (π−x), findf′(0).


28. Find limx→∞
    x− 25
    10 +x− 2 x^2

.

29. (Calculator) Letfbe a continuous and
    differentiable function. Selected values of

fare shown below. Find the approximate

value off′atx=2.

x 0 1 2 3 4 5

f 3.9 4 4.8 6.5 8.9 11.8

30. (Calculator) Iff(x)=

⎧

⎨

⎩

x^2 − 9

x− 3

, x=3,

3, x= 3

determine iff(x) is continuous at (x=3).

Explain why or why not.

7.11 Solutions to Practice Problems

Part A The use of a calculator is not

allowed.

1. Applying the power rule,
   dy
   dx
   = 30 x^4 −1.


2. Rewritef(x)=

1

x

+

1

√ (^3) x 2 as
f(x)=x−^1 +x−^2 /^3. Differentiate:
f′(x)=−x−^2 −

2

3

x−^5 /^3 =−

1

x^2

−

2

33

√

x^5

.

3. Rewrite

y=

5 x^6 − 1

x^2

asy=

5 x^6

x^2

−

1

x^2

= 5 x^4 −x−^2.

Differentiate:

dy

dx

= 20 x^3 −(−2)x−^3 = 20 x^3 +

2

x^3

.

An alternate method is to differentiate

y=

5 x^6 − 1

x^2

directly, using the quotient rule.

4. Applying the quotient rule,

dy

dx

=

(2x)(5x^6 −1)−(30x^5 )(x^2 )

(5x^6 −1)^2

=

10 x^7 − 2 x− 30 x^7

(5x^6 −1)^2

=

− 20 x^7 − 2 x

(5x^6 −1)^2

=

− 2 x(10x^6 +1)

(5x^6 −1)^2

.

5. Applying the product rule,u=(3x−2)^5
   andv=(x^2 −1), and then the chain rule,

f′(x)=[5(3x−2)^4 (3)][x^2 −1]+[2x]

×[(3x−2)^5 ]

=15(x^2 −1)(3x−2)^4 + 2 x(3x−2)^5

=(3x−2)^4 [15(x^2 −1)+ 2 x(3x−2)]

=(3x−2)^4 [15x^2 − 15 + 6 x^2 − 4 x]

=(3x−2)^4 (21x^2 − 4 x−15).