Homework Homework Assignment #22 Read Section 4.2 Page 217, Exercises: 1 – 65 (EOO) Rogawski...

Homework

Homework Assignment #22 Read Section 4.2 Page 217, Exercises: 1 – 65 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Example, 17Use the Linear Approximation to estimate Δf = f (3.02) – f (3) for the given function.

1. f (x) = x2


2

2

2

3 3 9, 3 2 3 6

3.02 3 3.02 3 3 6 0.02 9 9.12

3.02 9.12

L x f a x a f a f x x f x x

f f

L f f

L

Example, Page 217Use the Linear Approximation to estimate Δf = f (3.02) – f (3) for the given function.

5. f (x) = e2x


2 2

2 3 2 3

2

3 403.429, 3 2 806.858

3.02 3 3.02 3 3

806.858 0.02 403.429 419.566

3.02 419.566

x xL x f a x a f a f x e f x e

f e f e

L f f

L

Example, Page 217Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error.

9.


1 , 8, 1f x x a x

1 , 8, 1

1 1 18

62 1 2 1 8

1 11 9 8 1 9 1 8 0.1623

6 6

Error 0.1623 0.16 0.004389 Error 0.004389

0.004389%Error 100% 2.705% %Error 2.705%

0.1623

f x x a x

f f x x f x fx

f f f

Example, Page 217Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error.

13.


cos , , 0.034

f x x a x

2sin

4 2

20.03 0.0212 0.0212

2

cos 0.03 cos 0.02094 4

Error 0.0209 0.0212 0.0003

0.0003%Error 100% 1.538% Error 0.0003,%Error 1.538%

0.0209

f f x x f x x f

f f

Example, Page 217Estimate the quantity using Linear Approximation and find the error using a calculator.

17.


1 14 416.5 16

314 4

34

1 14 4

1 14 4

10.5

41 1 1 1 1

16 0.5 0.0156254 4 8 2 64

16.5 16 0.015625

16.5 16 0.015445 Error 0.0001798

f x x f x x x

f f x x

Example, Page 217Estimate the quantity using Linear Approximation and find the error using a calculator.

21.


sin 0.023

sin cos 0.023 0 cos0 1

1 0.023 0.023

sin 0.023 sin 0 0.02299797

Error 0.023 0.02299797 Error 0.0000020277797

f x x f x x x f

f f x x

Example, Page 21725. The cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using Linear Approximation.


1 23 3

1 13 3

1 1 10.2 27 0.037

3 3 9

0.037 0.2 0.0074 27.2 27 0.0074

f x x f x x x f

f f x x

Example, Page 21729. The atmospheric pressure P (kPa) at altitudes h (km) for 11 ≤ h ≤ 15 is approximately P (h) = 128 e–0.157h.

(a) Use Linear Approximation to estimate the change in pressure at h = 20 when Δh = 0.5.

(b) Compute the actual change and compute the percentage error in the Linear Approximation.


0.157 0.157

0.157 20

128 20.096 0.5

20 20.096 0.8698

0.8698 0.5 0.4349 0.4349

h hP h e f x e x

f e

P P h h P kPa

0.157 20.5 0.157 2020.5 20 128 0.4183

0.4183 0.4349% 100% 3.976% % 3.976%

0.4183

P P e e

Error Error

Example, Page 21733. A stone tossed vertically in the air with initial velocity v ft/s reaches a maximum height h = v2/64 ft.

(a) Estimate Δh if v is increased from 25 to 26 ft/s.

(b) Estimate Δh if v is increased from 30 to 31 ft/s.


2 2525 0.7812564 32 32

0.78125 1 0.78125 0.78125

v vh v h v h

h h v v h ft

2530 0.937532

0.9375 1 0.9375 0.9375

h

h h v v h ft

Example, Page 21733. (c) In general, does a 1 ft/s increase in initial velocity cause a greater change in maximum height at low or high initial velocities? Explain.

Since the value of the derivative of the height function increases as initial velocity increase, the greater change in maximum height occurs at high initial velocities.


Example, Page 21737. Estimate the weight loss per mile of altitude gained for a 130-lb pilot. At which altitude would she weigh 129.5 lb?


2 2 2 3

2 3

2 22

2

2

2

3960 2 130 3960 3960 3960 0.065 lb/mi

130 3960 130 3960129.5 7920 15681600

129.53960

7920 60546.71815 0 7.367

W x wR x W x wR x

W W

x xx

x x x mi

Example, Page 217Find the linearization at x = a.

41.


cos sin , 0y x x a

2 2cos sin , 0 cos sin cos 2

0 cos 2 0 1

0 1 0 0 1 0 0

y x x a f x x x x

L x f a x a f a f

f L x x x

L x x


45.


1

2 21 , 0y x a

32 2

32

11 2

2 1

0 1, 0 0

0 0 1 1

1

xf x x x

x

f f

L x f a x a f a x

L x


49.


ln , 1xy e x a

11

ln

1 ln 11

1 0 0 0 0 1

1

xxe

f x e xx

eL x f a x a f a f e e

f e L x e x e x

L x e x

Example, Page 217Approximate using linearization and use a calculator to compute percentage error.

53.


17

1Let 16

21 1

16 16 16 482 16

1 116 4 16 4

8 8

117 17 16 4 4.125 17 4.125 17 4.12310...

84.12310 4.125

% Error 100% 0.0459% % Error 0.0459%4.12310

f x x f x ax

L x f a x a f a f f

L x x x

L L


57.


1327.001

1 23 3

13

3

1Let 27

31 1

, 27 , 27 27 33 9 27

1 127 3 27 3

27 27

127.001 27.001 27 3 3.000037 27.001 3.000037

273.000037 3.000037

% Error 100% 0.0% % Error 0.0%3.000037

f x x f x x a

L x f a x a f a f f

L x x x

L


61.


1cos 0.52

1

2

2

1

1cos Let 0

11

, 0 1, 021 0

1 02 2

0.52 0.52 1.571 1.051 cos 0.52 1.024

1.024 1.051% Error 100% 2.622% % Error 2.622%

1.024

f x x f x ax

L x f a x a f a f f

L x x x

L

Example, Page 217From Newton’s Laws, an object released at angle θ with initial velocity v ft/s, travels a total distance s = 1/32 v2 sin 2θ ft.

65. Estimate the change in distance s of the shot if the angle changes from 50º to 51º for v = 25 ft/s and v = 30 ft/s. Is the shot more sensitive to the angle when the velocity is large or small? Explain.

Work on next slide.


Example, Page 21765. Continued


2

22

2

2

1sin 2

32

2cos 2 cos 2 Let 50

32 16

25Let 25 cos100 1 6.783

16

30Let 30 cos100 1 9.768

16The shot is more sensitive to change in angle when the velocity

is large 30 2

s s a x s v

vs x v a

v s

v s

s s

5 .


Jon Rogawski

Calculus, ETFirst Edition

Chapter 4: Applications of the DerivativeSection 4.2: Extreme Values


The extreme values of a function f (x) on an interval I are the minimum and maximum values of f (x) for x I

If f (x) is continuous on [a, b], then f (x) has a minimum and a maximum value on [a, b].

f (c) is a local maximum if f (x) ≤ f (c) for all x in some openinterval around c. Similarly, f (c) is a local minimum if f (x) ≥ f (c) for all x in some open interval around c.

c is a critical point of the function f (x) if either f (c) = 0 or f (c) D.N.E.


A doctor could use a graph like the one in Figure 1 to determine the maximum concentration of a drug in a patient’s bloodstream and the time after dosing at which it occurs.

Maximum and minimum values are called extreme values or extrema.The process of finding extrema is optimization.


A discontinuous function may or may not have a min or max.A function defined on an open interval may or may not have a min or max.A continuous function defined on a closed interval will have both a min and a max.


Examples of the local min and max are seen in Figure 3.


As shown in Figure 4(A), the tangent line at a min or max is horizontal. But in Figure 4(B), there is not tangent at the min,leading to the following definition:


Find the critical points ofx3 – 9x2 + 24x – 10.


Find the critical points for f (x) =|x|.


The converse of Fermat’s Theorem, that if c is a critical point, then f (c) is a min or max, is not necessarily true, as illustrated in Figure 8.

Example, Page 227Find all critical points of the function.

6.


3 28f t t t

Example, Page 227Find all critical points of the function.

8.


24 1f t t t


Figure 9 illustrates a situationwhere f (6) is a maximum, but6 is not a critical point. But f (4) is a minimum and 4 is a critical point.


Find the maximum of f (x) = 1 – (x – 1)2/3 on[–1, 2].


Find the extreme values of f (x) = x2 – 8lnx on [1, 4].


Rolle’s Theorem is illustrated in Figure 13.

Example, Page 227Find the minimum and maximum values of the function on the given interval.

54.


2 13 , , 1

2x xy e e

Example, Page 22758. Find the critical points of f (x) = 2cos3x +3cos2x. check your answer against a graph of f (x).


Example, Page 22772. Bees build honeycomb structures out of cells with a hexagonal base and three-rhombus shaped faces on top as in Figure 20. the surface area of the cell is

where h, s, and θ are as indicated in the figure.

(a) Show that by finding the critical point of A(θ) for 0 ≤ θ ≤ π/2, assuming h and s are constant


236 3 csc cot

2A hs s

54.7

Example, Page 22772. (b) Confirm by graphing that the critical point indeed minimizes the surface area.


3 csc cotf

Homework

Homework Assignment #23 Read Section 4.3 Page 227, Exercises: 1 – 77 (EOO), skip 57,

69


Homework Homework Assignment #22 Read Section 4.2 Page 217, Exercises: 1 – 65 (EOO) Rogawski...

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