Homework Homework Assignment #10 Read Section 3.2 Page 124, Exercises: 1 – 69 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework

Homework Assignment #10 Read Section 3.2 Page 124, Exercises: 1 – 69 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 1241. Let f (x) = 3x2. Show that f (2+h) =3h2 + 12h + 12.

Then show that and compute f ′(2) by taking the limit as h → 0.


22 2 22 3 2 3 4 4 3 12 12f x x f h h h h h h

2 23 12

f h fh

h

22 22 2 3 12 12 3 2 3 12 3 12f h f h h h h h

h h h

Homework, Page 124Compute f ′(a) in two ways, using Equations (1) and (2)5.


23 4 2, 1f x x x a

0

2 2

0

2

0

2

0 0

1 11 lim

3 1 4 1 2 3 1 4 1 2lim

3 6 3 4 4 2 3 4 2lim

6 3 4lim lim 6 3 4 2

1 2

h

h

h

h h

f h ff

h

h h

hh h h

hh h h h

hf

Homework, Page 1245. Continued


2

1

2

1

2

1

1 1

1 3 1 4 1 2 3 4 2 1

11 lim

1

3 4 2 1lim

13 4 1lim

13 1 1

lim lim 3 1 3 1 1 21

1 2

x

x

x

x x

f

f x ff

x

x xx

x xx

x xx

xf

Homework, Page 124Refer to the function whose graph is shown.9. Estimate for h = 0.5 and h = –0.5. Are these numbers larger or smaller than f ′ (2)? Explain.

The value of f ′ (2) lies between these two values, as illustrated in Figure 9.


2 2f h fh

2 0.5 2 2 10.5 0.5

22 0.5 2 0.6 1

0.5 0.50.8

f f

f f

Homework, Page 124Refer to the function whose graph is shown.13. Which is larger, f ′ (5.5) or f ′ (6.5)?

The value of f ′ (5.5) is larger than the value of f ′ (5.5) because the slope of the curve is steeper at x = 5.5 than it is at x = 6.5


Homework, Page 124Use the limit definition to find the derivative of the linear function.17. g (x) = 9 – t.


0 0

0 0

9 9lim lim

9 9lim lim 1

1

h h

h h

g t h g t t h tg t

h ht h t h

h hg t

Homework, Page 12421. Let . Show that

Then use this formula to compute f ′(9) (by taking the limit).


9 9 19 3

f h fh h

f x x

0 0

9 9 9 9 9 3 9 39 3

9 9 19 39 3

9 9 1 19 lim lim69 3

196

h h

f h f h h hh h h h

hhh h

f h ff

h h

f

Homework, Page 124Compute the derivative at x = a using the limit definition and find an equation of the tangent line.25. f (x) = x3, a = 2


3 3 2 32

0 0 0

3

2 2 8 12 6 82 lim lim lim 12 6

12 2 12

2 2 8 8 12 2

h h h

h h h hf h hh h

f

f y x

Homework, Page 124Compute the derivative at x = a using the limit definition and find an equation of the tangent line.29. f (x) = x–1, a = 3


1 1

0 0 0

0 0 0

1

1 13 3 3 3 3 33 33 lim lim lim

3 3 3 3

3 3 1 1 1lim lim lim 33 3 3 3 3 3 9 9

1 1 13 3 33 3 9

h h h

h h h

h h hhfh h h h h

h h fh h h h h

f y x

Homework, Page 124Compute the derivative at x = a using the limit definition and find an equation of the tangent line.33.


1 , 23

f x ax

0 0

0 0 0

0

1 1 1 12 3 2 3 11 12 lim lim

11 1 1 1lim lim lim

1 1 1

1lim 1 2 11

12 1 1 1 2 1 22 3

h h

h h h

h

h hhfh h h

h h hh h h h h h

fh

f y x y x

Homework, Page 124Compute the derivative at x = a using the limit definition and find an equation of the tangent line.37.


3 , 1f t t a

3 3 2 32 3

2 30 0

2 3 2 3

2 3 2 30 0

2

2 30

3

1 1 1 11 1 1 3 31 3 3 11 lim lim

1 3 31 1 3 3 3 3lim lim

1 3 3 1 3 3

3 3lim 3 1 31 3 3

11 1 1 3 11

h h

h h

h

h h h hh h hfh h h h h

h h h h h hh h h h h h h h

h h fh h h

f y x

Homework, Page 12441. What is the equation of the tangent line at x = 3, assuming that f (3) = 5 and f ′(3) = 2?


5 2 3y x

Homework, Page 12445.

f ′(1) appears to be about – 0.5


2

1 1 Verify that 1, lies on the graphs of both 2 1

and 1 for every slope . Plot and on the

same axes for several values of . Experiment until you find a value of fo

P f xx

L x m x m f x L x

mm

r which appears tangent to the graph of .

What is your estimate of 1 ?

y L x f x

f

x

y

m = -2

m = -1

m = -0.5

Homework, Page 12449. The vapor pressure of water is defined as the atmospheric pressure P at which no net evaporation takes place. See graph and table.

(a) Which is larger, P′ (300) or P′ (350)? From the graph, P′ (300) < P′ (350) (b) Estimate P′ (T) for T = 303, 313, 323, 333, 343 using the table and the average of the difference quotients for h = 10 and h = –10.


0.25

0.5

0.75

1.0

293 313 333 353 373

T (K) P (atm) T (K) P (atm)293 0.0278 333 0.2067

303 0.0482 343 0.3173

313 0.0808 353 0.4754

323 0.1311

Homework, Page 12449. (b) Estimate P′ (T) for T = 303, 313, 323, 333, 343 using the table and the average of the difference quotients for h = 10 and h = –10.


303 10 303 10 0.0808 0.0278303 0.0026520 20

313 10 313 10 0.1311 0.0428313 0.0044220 20

323 10 323 10 0.2067 0.0808323 0.0063020 20

333 10 333 10 0.3173 0.1311333 0.0093120 20

343 10 34343

P PP

P PP

P PP

P PP

P PP

3 10 0.4754 0.2067 0.0134420 20

Homework, Page 124Each of the limits represents a derivative f '(a). Find f (x) and a.


3

0

5 12553. lim

h

hh

3

0 0

3

5 125lim lim

, 5

h h

h f a h f ah h

f x x a

Homework, Page 124Each of the limits represents a derivative f '(a). Find f (x) and a.


2

0

5 2557. limh

h h

2

0 0

5 25lim lim

5 , 2

h

h h

x

f a h f ah h

f x a

Homework, Page 12461. Let (a) Plot f (x) over [1, –1]. Then zoom in near x = 0 until the graph

appears straight and estimate the slope f '(x).The slope appears to be about –0.1.

(b) Use your estimate to find an approximate equation to the tangent line at x = 0. Plot this line and the graph on the same axes.


41 2xf x

0.1 2y x

Homework, Page 12465. Apply the method of Example 6 to f (x) = sin x to determine accurately to four decimal places.

The difference quotient yields the value of accurate to four decimal places with h = ± 0.00001. The value according to our calculators is 0.7071067812……


4f

sin 4

h > 0 Diff Quot h < 0 Diff Quot0.01 0.70356 -0.01 0.71063

0.001 0.70675 -0.001 0.70746

0.0001 0.70707 -0.0001 0.70714

0.00001 0.70710 -0.00001 0.70711

Homework, Page 12469. Figure 18 gives the average antler weight W of a red deer as a function of age t. Estimate the slope of the tangent line to the graph at t = 4. For which values of t is the slope of the tangent line equal to zero? For which is it negative?

The slope of the tangent appearsto equal zero at t = 10 and t = 11.5. The slope is negative on [10, 11.5].


7 2 548 2 6

m


Jon Rogawski

Calculus, ET First Edition

Chapter 3: DifferentiationSection 3.2: The Derivative as a

Function

0

We have previously used the definition

lim

to find the slope of a curve at a particular point. h

f a h f af a

h

The Derivative of a Function

If we substitute for in the definition, we obtain a general formula for the derivative of a function, i.e.,

x a

0

lim . h

f x h f xf x

h

The derivative of a function is also a function.

The Derivative as a Function

If y = f (x), we may also write y′ or y' (x) for f '(x). The domain of f '(x) consists of all values of x in the

domain of f (x) for which the limit exists. Function f (x) is said to be differentiable on (a, b) if

f '(x) exists for all x in (a, b). If f '(x) exists for all values of x in the domain of f

(x), then we say f (x) is differentiable.

Use the definition to find the derivative of .f x x

Example:Finding the Derivative of a Function


3

Use the definition to find the derivative

of 12 and verify the equation ofthe tangent line at 3.

y x xx

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Homework Homework Assignment #10 Read Section 3.2 Page 124, Exercises: 1 – 69 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company