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

MA 3972-MA-Book April 11, 2018 15:14

More Applications of Derivatives 221

23. Evaluate limx→ 100
    √x−^100
    x− 10

.

24. A functionfis continuous on the interval
    (−1, 8) withf(0)=0,f(2)=3, and
    f(8)= 1 /2 and has the following
    properties:

INTERVALS (−1, 2) x=2 (2, 5) x=5 (5, 8)

f′ + 0 −−−

f′′ −−− 0 +

(a) Find the intervals on which fis

increasing or decreasing.

(b) Find wheref has its absolute extrema.

(c) Find wheref has the points of

inflection.

(d) Find the intervals on whichfis

concave upward or downward.

(e) Sketch a possible graph of f.

25. The graph of the velocity function of a
    moving particle for 0≤t≤8 is shown in
    Figure 10.6-1. Using the graph:

(a) Estimate the acceleration when

v(t)=3 ft/sec.

(b) Find the time when the acceleration is

a minimum.

8 7 6 5 4 3 2 1

0123456789

t

v

v(t)

Feet/sec

Seconds

Figure 10.6-1

10.7 Solutions to Practice Problems

Part A The use of a calculator is not

allowed.

1. Equation of tangent line:
   y=f(a)+f′(a)(x−a)
   f′(x)=

1

4

( 1 +x)−^3 /^4 (1)=

1

4

( 1 +x)−^3 /^4

f′(0)=

1

4

and f(0)=1;

thus,y= 1 +

1

4

(x− 0 )= 1 +

1

4

x.

f(0.1)= 1 +

1

4

(0.1)= 1. 025

2. f(a+Δx)≈ f(a)+ f′(a)Δx

Letf(x)=^3

√

xand f( 28 )=f( 27 + 1 ).

Thenf′(x)=

1

3

(x)−^2 /^3 ,

f′( 27 )=

1

27

, and f( 27 )=3.

f( 27 + 1 )≈f( 27 )+f′( 27 )( 1 )≈

3 +

(

1

27

)

( 1 )≈ 3. 037

3. f(a+Δx)≈ f(a)+f′(a)Δx
   Convert to radians:
   46
   180

=

a

π

⇒a=

23 π

90

and 1◦=

π

180

;

45 ◦=

π

4

.

Let f(x)=cosxandf(45◦)=

f

(

π

4

)

=cos

(

π

4

)

=

√

2

2

.