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

(Marvins-Underground-K-12) #1
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.



  1. Find lim
    h→ 0


sin

(
π
2
+h

)
−sin

(
π
2

)

h

.



  1. Iff(x)=cos^2 (π−x), findf′(0).

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


.



  1. (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


  1. (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

.



  1. 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.


  1. 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

.



  1. 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).
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