98 STEP 4. Review the Knowledge You Need to Score High
6.9 Cumulative Review Problems
(Calculator) indicates that calculators are
permitted.- Find limh→ 0
sin(
π
2
+h)
−sin(
π
2)h.
- Iff(x)=cos^2 (π−x), findf′(0).
- Find limx→∞
x− 25
10 +x− 2 x^2
.
- (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- (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.6.10 Solutions to Practice Problems
Part A The use of a calculator is not
allowed.- Applying the power rule,
dy
dx
= 30 x^4 −1. - 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.
- 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 differentiatey=
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).