 Precalculus 2e

Chapter 12

Precalculus 2eChapter 12

12.1Finding Limits: Numerical and Graphical Approaches

1 .

$a=5, a=5,$ $f( x )=2 x 2 −4, f( x )=2 x 2 −4,$ and $L=46. L=46.$

2 .

a. 0; b. 2; c. does not exist; d. $−2; −2;$ e. 0; f. does not exist; g. 4; h. 4; i. 4

3 .

$lim x→0 ( 20sin(x) 4x )=5 lim x→0 ( 20sin(x) 4x )=5$ 4 .

does not exist

12.2Finding Limits: Properties of Limits

1 .

26

2 .

59

3 .

10

4 .

$−64 −64$

5 .

$−3 −3$

6 .

$− 1 50 − 1 50$

7 .

$− 1 8 − 1 8$

8 .

$2 3 2 3$

9 .

$−1 −1$

12.3Continuity

1 .
1. removable discontinuity at $x=6; x=6;$
2. jump discontinuity at $x=4 x=4$
2 .

No. The function is not continuous at $x=2x=2$ because the left hand limit is $1212$ and the right hand limit is $6.56.5$.

3 .

No, the function is not continuous at $x=3. x=3.$ There exists a removable discontinuity at $x=3. x=3.$

4 .

$x=6 x=6$

12.4Derivatives

1 .

3

2 .

$f ′ (a)=6a+7 f ′ (a)=6a+7$

3 .

$f ′ (a)= −15 ( 5a+4 ) 2 f ′ (a)= −15 ( 5a+4 ) 2$

4 .

$3 2 3 2$

5 .

0

6 .

$−2 , −2 ,$ 0, 0, $−3 −3$

7 .
1. After zero seconds, she has traveled 0 feet.
2. After 10 seconds, she has traveled 150 feet east.
3. After 10 seconds, she is moving eastward at a rate of 15 ft/sec.
4. After 20 seconds, she is moving westward at a rate of 10 ft/sec.
5. After 40 seconds, she is 100 feet westward of her starting point.
8 .

The graph of $f f$ is continuous on $( −∞,1 )∪( 1,3 )∪( 3,∞ ). ( −∞,1 )∪( 1,3 )∪( 3,∞ ).$ The graph of $f f$ is discontinuous at $x=1 x=1$ and $x=3. x=3.$ The graph of $f f$ is differentiable on $( −∞,1 )∪( 1,3 )∪( 3,∞ ). ( −∞,1 )∪( 1,3 )∪( 3,∞ ).$ The graph of $f f$ is not differentiable at $x=1 x=1$ and $x=3. x=3.$

9 .

$y=19x−16 y=19x−16$

10 .

–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

12.1 Section Exercises

1 .

The value of the function, the output, at $x=a x=a$ is $f( a ). f( a ).$ When the $lim x→a f( x ) lim x→a f( x )$ is taken, the values of $x x$ get infinitely close to $a a$ but never equal $a. a.$ As the values of $x x$ approach $a a$ from the left and right, the limit is the value that the function is approaching.

3 .

–4

5 .

–4

7 .

2

9 .

does not exist

11 .

4

13 .

does not exist

15 .

16 .

17 .

18 .

19 .

20 .

21 .

23 .

7.38906

25 .

54.59815

27 .

$e 6 ≈403.428794, e 6 ≈403.428794,$ $e 7 ≈1096.633158, e 7 ≈1096.633158,$ $e n e n$

29 .

$lim x→−2 f(x)=1 lim x→−2 f(x)=1$

31 .

$lim x→3 ( x 2 −x−6 x 2 −9 )= 5 6 ≈0.83 lim x→3 ( x 2 −x−6 x 2 −9 )= 5 6 ≈0.83$

33 .

$lim x→1 ( x 2 −1 x 2 −3x+2 )=−2.00 lim x→1 ( x 2 −1 x 2 −3x+2 )=−2.00$

35 .

$lim x→1 ( 10−10 x 2 x 2 −3x+2 )=20.00 lim x→1 ( 10−10 x 2 x 2 −3x+2 )=20.00$

37 .

$lim x→ −1 2 ( x 4 x 2 +4x+1 ) lim x→ −1 2 ( x 4 x 2 +4x+1 )$ does not exist. Function values decrease without bound as $x x$ approaches –0.5 from either left or right.

39 .

$lim x→0 7tanx 3x = 7 3 lim x→0 7tanx 3x = 7 3$ 40 . 43 .

$lim x→0 e e − 1 x 2 =1.0 lim x→0 e e − 1 x 2 =1.0$

45 .

$lim x→− 1 − | x+1 | x+1 = −(x+1) (x+1) =−1 lim x→− 1 − | x+1 | x+1 = −(x+1) (x+1) =−1$ and $lim x→− 1 + | x+1 | x+1 = (x+1) (x+1) =1; lim x→− 1 + | x+1 | x+1 = (x+1) (x+1) =1;$ since the right-hand limit does not equal the left-hand limit, $lim x→−1 | x+1 | x+1 lim x→−1 | x+1 | x+1$ does not exist.

47 .

$lim x→−1 1 ( x+1 ) 2 lim x→−1 1 ( x+1 ) 2$ does not exist. The function increases without bound as $x x$ approaches $−1 −1$ from either side.

49 .

$lim x→0 5 1− e 2 x lim x→0 5 1− e 2 x$ does not exist. Function values approach 5 from the left and approach 0 from the right.

51 .

Through examination of the postulates and an understanding of relativistic physics, as $v→c, v→c,$ $m→∞. m→∞.$ Take this one step further to the solution,

$lim v→ c − m= lim v→ c − m o 1−( v 2 / c 2 ) =∞ lim v→ c − m= lim v→ c − m o 1−( v 2 / c 2 ) =∞$

12.2 Section Exercises

1 .

If $f f$ is a polynomial function, the limit of a polynomial function as $x x$ approaches $a a$ will always be $f( a ). f( a ).$

3 .

It could mean either (1) the values of the function increase or decrease without bound as $x x$ approaches $c, c,$ or (2) the left and right-hand limits are not equal.

5 .

$−10 3 −10 3$

7 .

6

9 .

$1 2 1 2$

11 .

6

13 .

does not exist

15 .

$−12 −12$

17 .

$− 5 10 − 5 10$

19 .

$−108 −108$

21 .

1

23 .

6

25 .

1

27 .

1

29 .

does not exist

31 .

$6+ 5 6+ 5$

33 .

$3 5 3 5$

35 .

0

37 .

$−3 −3$

39 .

does not exist; right-hand limit is not the same as the left-hand limit.

41 .

2

43 .

Limit does not exist; limit approaches infinity.

45 .

$4x+2h 4x+2h$

47 .

$2x+h+4 2x+h+4$

49 .

$cos(x+h)−cos(x) h cos(x+h)−cos(x) h$

51 .

$−1 x(x+h) −1 x(x+h)$

53 .

$−1 x+h + x −1 x+h + x$

55 .

$f( x )= x 2 +5x+6 x+3 f( x )= x 2 +5x+6 x+3$

57 .

does not exist

59 .

52

12.3 Section Exercises

1 .

Informally, if a function is continuous at $x=c , x=c ,$ then there is no break in the graph of the function at $f( c ), f( c ),$ and $f( c ) f( c )$ is defined.

3 .

discontinuous at $a=−3 a=−3$; $f(−3) f(−3)$ does not exist

5 .

removable discontinuity at $a=−4 a=−4$; $f(−4) f(−4)$ is not defined

7 .

Discontinuous at $a=3 a=3$; $lim x→3 f(x)=3 , lim x→3 f(x)=3 ,$ but $f(3)=6 , f(3)=6 ,$ which is not equal to the limit.

9 .

$lim x→2 f(x) lim x→2 f(x)$ does not exist.

11 .

$lim x→ 1 − f(x)=4; lim x→ 1 + f(x)=1 lim x→ 1 − f(x)=4; lim x→ 1 + f(x)=1$. Therefore, $lim x→1 f(x) lim x→1 f(x)$ does not exist.

13 .

$lim x→ 1 − f(x)=5≠ lim x→ 1 + f(x)=−1 lim x→ 1 − f(x)=5≠ lim x→ 1 + f(x)=−1$. Thus $lim x→1 f(x) lim x→1 f(x)$ does not exist.

15 .

$lim x→− 3 − f(x)=−6 lim x→− 3 − f(x)=−6$, $lim x→− 3 + f(x)=− 1 3 lim x→− 3 + f(x)=− 1 3$

Therefore, $lim x→−3 f(x) lim x→−3 f(x)$ does not exist.

17 .

$f( 2 ) f( 2 )$ is not defined.

19 .

$f( −3 ) f( −3 )$ is not defined.

21 .

$f( 0 ) f( 0 )$ is not defined.

23 .

Continuous on $(−∞,∞) (−∞,∞)$

25 .

Continuous on $(−∞,∞) (−∞,∞)$

27 .

Discontinuous at $x=0 x=0$ and $x=2 x=2$

29 .

Discontinuous at $x=0 x=0$

31 .

Continuous on $(0,∞) (0,∞)$

33 .

Continuous on $[4,∞) [4,∞)$

35 .

Continuous on $(−∞,∞) (−∞,∞)$.

37 .

1, but not 2 or 3

39 .

1 and 2, but not 3

41 .

$f( 0 ) f( 0 )$ is undefined.

43 .

$(−∞,0)∪(0,∞) (−∞,0)∪(0,∞)$

45 .

At $x=−1, x=−1,$ the limit does not exist. At $x=1, x=1,$ $f( 1 ) f( 1 )$ does not exist.

At $x=2, x=2,$ there appears to be a vertical asymptote, and the limit does not exist.

47 .

$x 3 +6 x 2 −7x ( x+7 )( x−1 ) x 3 +6 x 2 −7x ( x+7 )( x−1 )$

49 .

The function is discontinuous at $x=1 x=1$ because the limit as $x x$ approaches 1 is 5 and $f( 1 )=2. f( 1 )=2.$

12.4 Section Exercises

1 .

The slope of a linear function stays the same. The derivative of a general function varies according to $x. x.$ Both the slope of a line and the derivative at a point measure the rate of change of the function.

3 .

Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.

5 .

The average rate of change of the amount of water in the tank is 45 gallons per minute. If $f( x ) f( x )$ is the function giving the amount of water in the tank at any time $t ,t,$ then the average rate of change of $f( x ) f( x )$ between $t=a t=a$ and $t=b t=b$ is $f(a)+45(b−a). f(a)+45(b−a).$

7 .

$f ′ (x)=−2 f ′ (x)=−2$

9 .

$f ′ (x)=4x+1 f ′ (x)=4x+1$

11 .

$f ′ (x)= 1 (x−2) 2 f ′ (x)= 1 (x−2) 2$

13 .

$−16 ( 3+2x ) 2 −16 ( 3+2x ) 2$

15 .

$f ′ (x)=9 x 2 −2x+2 f ′ (x)=9 x 2 −2x+2$

17 .

$f ′ (x)=0 f ′ (x)=0$

19 .

$− 1 3 − 1 3$

21 .

undefined

23 .

$f ′ (x)=−6x−7 f ′ (x)=−6x−7$

25 .

$f ′ (x)=9 x 2 +4x+1 f ′ (x)=9 x 2 +4x+1$

27 .

$y=12x−15 y=12x−15$

29 .

$k=−10 k=−10$ or $k=2 k=2$

31 .

Discontinuous at $x=−2 x=−2$ and $x=0. x=0.$ Not differentiable at –2, 0, 2.

33 .

Discontinuous at $x=5. x=5.$ Not differentiable at -4, –2, 0, 1, 3, 4, 5.

35 .

$f( 0 )=−2 f( 0 )=−2$

37 .

$f( 2 )=−6 f( 2 )=−6$

39 .

$f ′ ( −1 )=9 f ′ ( −1 )=9$

41 .

$f ′ ( 1 )=−3 f ′ ( 1 )=−3$

43 .

$f ′ ( 3 )=9 f ′ ( 3 )=9$

45 .

Answers vary. The slope of the tangent line near $x=1 x=1$ is 2.

47 .

At 12:30 p.m., the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

49 .

At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

51 .

The height of the projectile after 2 seconds is 96 feet.

53 .

The height of the projectile at $t=3 t=3$ seconds is 96 feet.

55 .

The height of the projectile is zero at $t=0 t=0$ and again at $t=5. t=5.$ In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

57 .

$36π 36π$

59 .

$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold. 61 .$21 per unit

63 .

\$36

65 .

$f'(x)=10a−1 f'(x)=10a−1$

67 .

$4 ( 3−x ) 2 4 ( 3−x ) 2$

Review Exercises

1 .

2

3 .

does not exist

5 .

Discontinuous at $x=−1( lim x→a f(x)x=−1( lim x→a f(x)$ does not exist $),x=3(),x=3($jump discontinuity$))$, and$x=7( lim x→a f(x)x=7( lim x→a f(x)$ does not exist$). ).$

7 .

$lim x→−2 f(x)=0 lim x→−2 f(x)=0$

9 .

Does not exist

11 .

$-35-35$

13 .

$11$

15 .

$66$

17 .

$500500$

19 .

$-67-67$

21 .

At $x=4, x=4,$ the function has a vertical asymptote.

23 .

At $x=3, x=3,$ the function has a vertical asymptote.

25 .

Removable discontinuity at $a=9a=9$

27 .

Removable discontinuity at $x=5x=5$

29 .

Removable discontinuity at $x=5x=5$, discontinuity at $x=1x=1$

31 .

Removable discontinuity at $x=-2x=-2$, discontinuity at $x=5x=5$

33 .

$33$

35 .

$1 (x+ 1)(x+h+1) 1 (x+ 1)(x+h+1)$

37 .

$e2x+2h-e2x h e2x+2h-e2x h$

39 .

$10x-3 10x-3$

41 .

The function would not be differentiable at however, 0 is not in its domain. So it is differentiable everywhere in its domain.

Practice Test

1 .

3

3 .

0

5 .

$−1 −1$

7 .

$lim x→ 2 − f(x)=− 5 2 a lim x→ 2 − f(x)=− 5 2 a$ and $lim x→ 2 + f(x)=9 lim x→ 2 + f(x)=9$ Thus, the limit of the function as $x x$ approaches 2 does not exist.

10 .

$-150 -150$

12 .

$1 1$

14 .

Removable discontinuity at $x=3 x=3$

16 .

$f'(x)=- 3 2a32 f'(x)=- 3 2a32$

18 .

Discontinuous at −2, 0, not differentiable at −2, 0, 2

20 .

Not differentiable at $x=0 x=0$ (no limit)

22 .

The height of the projectile at $t=2 t=2$ Seconds

24 .

The average velocity from $t=1 t=1$ $t=2 t=2$

26 .

$13 13$

28 .

$0 0$

29 .

2

30 .

$x=1 x=1$

32 .

$y=−14x−18 y=−14x−18$

34 .

The graph is not differentiable at $x=1 x=1$ (cusp).

36 .

$f ' (x)=8x f ' (x)=8x$

38 .

$f ' (x)=− 1 ( 2+x ) 2 f ' (x)=− 1 ( 2+x ) 2$

40 .

$f ' (x)=−3 x 2 f ' (x)=−3 x 2$

42 .

$f'(x)= 1 2 x−1 f'(x)= 1 2 x−1$