149

Lesson 6-1 Slope Fields, Euler’s Method

A slope field is a graphical representation of a set of slopes obtained from a differential equation.

Remember that a differential equation involves a derivative. That derivative represents the slopes for a

function. Even if you cannot separate variables and integrate, you can still use a differential equation to

plot the slopes for a function.

Example 1: Find the slopes given by the differential equation dy x

dx y

at the following points:

a. (3, 2) b. ( 1, 3) c. ( 2, 1) d. (2, 2)

Why can’t you find slopes when y = 0 ?

Example 2: Find and plot the slopes given

by dy x

dx y

for each remaining marked

point (dot) in the coordinate plane at the right.

Example 3: In Example 2, you made what is known as a slope field. Starting at the point (0, 1),

follow the flow of the slopes to sketch the solution curve containing (0, 1). Your graph should be

“parallel” to the slope lines and be like an “average of slopes” whenever it goes between lines. Your

solution curve must represent a function whose domain is the largest possible open interval containing

the given point. Sketch a solution curve passing through ( 1,1) and one passing through (0, 3) .

Note: The most common student error in sketching a particular solution to a differential equation is to

extend the sketch too far and create a graph which is not a function.

Example 4: Solve the differential equation xdy

dx y

.

Note: Solving for y yields y ________________ or y ________________ .

Find the particular solution for this differential equation whose graph passes through the point (0,1) .

Find the particular solution whose graph passes through the point (0, 3) .

150

Example 5: For the differential equation 1

yy

a. Draw the slope field in the dot coordinate plane at the right.

b. Graph the particular solutions passing through the

points ( 2, 1) and (2,2) as functions of x.

c. Solve the differential equation.

d. Write as functions the particular solutions for the differential equation

whose graphs pass through ( 2, 1) and (2,2) .

Example 6: Which of the differential equations

below matches the slope field shown at the right?

a. y x b. y y c. y x y

d. 21y y e. 21y x

Example 7: The slope field for a certain differential

equation is shown at the right. Which of the following

could be a specific solution to the differential equation?

a. xy e b. xy e c. xy e

d. lny x e. lny x

x

y

x

y

151

Euler’s Method This is a more precise method of graphing an approximate solution to a differential

equation.

Example 8. Use Euler’s method to construct an Example 9. Solve dy

dxy

approximate solution for the differential equation algebraically. Fill in the table dy

dxy . Start at the point 0,1 and use step size with the actual values of y.

.1x

.3y

Example 10. Use Euler’s Method to approximate the particular solution of the diff. eq. y x y

passing through the point 0,0.5 . Let .2x and do three steps (n = 3). Graph the points.

.6y

Example 11. Sketch a particular solution of the diff. eq.

y x y passing through the point 0,0.5 using the

slope field given. Do the two graphs coincide?

x

y

x

y

x

y

0 1

.1

.2

.3

x y

slopedy

dxx y y x

slopedy

dxx y y x

152

Assignment 6-1

1. Find the slopes given by the differential equation 2

2

xy

y

at each of the following points:

a. (0,0) b. (1,1) c. ( 2,4) d. (4, 2) e. ( 3, 3) f. (5,12)

2. For the differential equation in Problem 1, why are there no slopes when y = 2 ?

3. The slope field for x

yy

is shown at the right.

a. Plot the following points on the slope field:

i. (1, 2) ii. (3, 1) iii. (0, 3)

iv. (0, 2) v. ( 2, 1)

b. Plot a separate solution curve through each of

the points from Part a. Remember that the

curves have to be functions.

c. What would a solution curve containing

(2, 2) look like?

d. Solve the differential equation x

yy

.

4. For the differential equation dy

dxy

a. Draw the slope field for the differential equation.

b. Graph the particular solutions passing

through the points (0,1) and (0, 1) .

c. Solve the differential equation, and find

the particular solutions that contain the

points (0, 1) and (0, 1) .

5. For the differential equation 2

dy

dx

x

a. Draw the slope field for the differential equation.

b. Graph the particular solution passing through the point ( 1,1) .

c. Solve the differential equation, and find the particular solution that contains

the point ( 1,1) .

6. Repeat the three parts of Problem 5 for the differential equation 1

2y

y . For this problem, draw

your graph as and write your solution as a function of x.

7. Repeat the three parts of Problem 5 for the differential equation 2 1y y .

8. Repeat the three parts of Problem 5 for the differential equation 2 ( 1)dy

dxx y , but use the

origin (instead of ( 1,1) ) for Parts b. and c.

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

x

y

2

2

2

2

x

y

153

Assignment 6-1 tear-out page Name _________________________ Per ____

1. a. ____ b. ____ c. ____ d. ____ e. ____ f. ____ 2.

3. a, b, c 3. d. solve x

yy

4. a, b 4. c.

5. a, b 5. c.

6. a, b 6. c.

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

. . . . . . . m=

x

y

2

22

2

154

7. a, b 7. c.

8. a, b 8. c.

9. ____ 10. ____

155

9. The slope field for a certain differential equation is

shown at the right. Which of the following could be

a specific solution to that differential equation.

a. 31

32x y b.

2 2 4x y

c. 2 2 4x y d. 4

yx

e. 4y

x

10. Which of the differential equations below

matches the slope field shown at the right?

a. dy

dxx y b.

dy

dxy x

c. dy

dx

x

y d.

dy

dx

y

x e.

dy

dxxy

The differential equation 3 34x y x y cannot be solved using the method of separation of variables.

Use substitution to determine if the function given is a solution of this differential equation.

11. sin 2y x 12. 2xy e

13. Without using a calculator, use Euler’s Method with a step size of 0.1 to approximate

.3f if 0 3f

and .f x x y

14. Without using a calculator, use Euler’s Method with 3 steps each with a size of 1

2 to

approximate a y-value if 0 2y and 2 3 .y x y

15. Using a calculator, if 1 2y and xyy e use 4 steps of Euler’s Method to approximate

0.8 .y

16. If 0

( )( ) lim

h

t h t h t tf t

h

, find ( )f t

Find the x-value(s) where each of the functions in Problems 17-21 is not differentiable. Give a reason

why each function is not differentiable for those values of x.

17. 2( ) 9f x x 18. ( ) 2( 1)g x x 19.

2 ( 2)( )

( 1)

x xp x

x x

20. 1

3( )q x x x 21. 21 3

2 2

2 1, 1( )

, 1

x xh x

x x x

x

y

x

y

156

22. If a particle moves along the curve 2

3y x , such that 3dx

dt for all x, find:

a. when 1dy

dtx b. when 8

dy

dtx c. lim

x

dy

dt d.

0limx

dy

dt

23. a. Use a tangent line to the graph of 2

3y x to approximate 2

3(8.1) .

b. Why could a tangent line to 2

3y x at 0x not be used to approximate 2

3(.1) ?

24. Without using a calculator, find the domain, x- and y-intercepts, asymptotes, and holes for 3

3

2 2

4

x xy

x x

. (Remember to factor, reduce, and use the reduced function for everything except

domain restrictions.) Then, sketch the graph of

3

3

2 2

4

x xy

x x

without using a calculator. Check

your graph with a calculator.

25. A region in Quadrant 1 is enclosed by the graphs of , cos ,y x y x and the y-axis. Sketch the

region, draw a representative rectangle, set up an integral, and find the volume of the solid for

each revolution described. Use a calculator, and make sure that it is radian mode.

a. about the x-axis. b. about y = 1 c. about the y-axis d. about 1x

Evaluate in Problems 26-31 without using a calculator.

26. 2

0

1

5 2dt

t 27. 3

2

1

1

ydy

y

28.

2

3 10

xdx

x

29. 3(ln )

du

u u 30. 25 x dx 31. 25 dx

32. Sketch a graph of a function ( )f x having the following characteristics:

f is continuous. ( 2) (0) 0f f .

(0)f is undefined. ( ) 0 for 1 and for 0f x x x . ( ) 0 on ( 1,0)f x .

( ) 0 for 0, and ( ) 0 for 0f x x f x x

33. Find the equation of a tangent line to 2 23 3 at (1,2).x yx y

34. A radioactive element has a half-life of 1000 years. How much of 200 grams of the element

will remain after 750 years?

157

Selected Answers

1a. 0 c. 2 e. 9

5 3b. i,ii.

3d. 2 2 or y x C y x C

4c. xy e for 0,1

5a,b. c. 21 3

4 4y x

6c. 2, 2y x x 7c. 3 4

33 4,y x x 8c.

2

1xy e

11. not a solution 13. 0.3 4.024f 14. 13

22y 15. 0.8 1.078 or 1.079y

17. 3x (sharp turns) 19. 0x (hole), 1x (VA) 20. 0x (vertical tangent)

22a. 2 c. 0 23a. 1

304 24. Do: 0, 2x ,

2 1 1

2 2red

x xy

x x

, VA: 2x (odd),

Hole: 1

20, , x-int: 1,0 (odd), no y-int., HA: 2y 25a. 1.520 b. 4.036 or 4.037

25c. .643 d. 3.159 or 3.160 26. ln 5 27. 1 2ln 2 29. 21

2lnu C

31. 25x C 33. 2 8 1y x 34. 118.920 or 118.921 grams

Lesson 6-2 Inverse Trig Functions

Definition of the Inverse Trig Functions:

Function Domain ( x values) *Range ( y values)

arcsin siny x y x 1,1

arccos cosy x y x [ 1,1] [ , ]0

arctan tany x y x ( , )

x

y

x

y

158

Example 1: Graph the indicated inverse trig functions in the coordinate planes below:

arcsiny x arccosy x arctany x

*A geometric representation of the range values

for each inverse trig function is shown in the arccos

coordinate plane at right.

arcsin

arctan

Example2: Evaluate without a calculator.

a. arctan1 b. 1cos ( 1) c. 1 3

2sin

d. arcsin 2

Example 3: Solve for x. 2

2arcsin( 3)x

Examples: Sketch a right triangle, and evaluate without a calculator.

4. Find tan x , given that 2

5arccosx 5. Find cos y , given that arcsiny x

Example 6: Use your answer from Example 5 to find arcsind

dxx .

x

y

x

y

x

y

0

2

2

159

Derivatives of the Inverse Trig Functions:

2

1arcsin

1

d

dxx

x

2arcsin

1

d

dx

uu

u

2

1arccos

1

d

dxx

x

2arccos

1

d

dx

uu

u

2

1arctan

1

d

dxx

x

2arctan

1

d

dx

uu

u

(where u is a function of x)

Examples: Differentiate.

7. ( ) arctan(2 1)g y y 8. ( ) arcsinf x x

9. ( ) arccos lnh t t 10. 2

arcsinx

y simplify

Integration of Inverse Trig Functions

1. 2 2

arcsinu u

dx Caa u

2.

2 2

1arctan

u udx C

a u a a

Examples: Integrate.

11. 2

1

4dx

x 12.

24 25

dx

x

160

13. 2

8

3 4dx

x 14. 2

8

3 4

xdx

x

15.

2

2

8

3 4

xdx

x 16. 2

4

4

xdx

x

Assignment 6-2

Evaluate the expressions in Problems 1-4 without using a calculator.

1. 3

2arcsin 2. arctan( 1) 3. 1

2arccos

4. arctan 3

In Problems 5 and 6, solve for x without using a calculator.

5. 4

arctan(3 )x

6. 2arccos 2x

For Problems 7 and 8, evaluate without using a calculator. First, sketch a triangle for each problem.

7. Find cos y , given that 4

5arcsiny

. 8. Find sin x , given that arctan(3)x .

Differentiate in Problems 9-14. Do not use a calculator.

9. 2arctan(3 )y x 10. 2( ) arcsin 1f x x 11. ( ) arcsin yg y e

12. 32( ) arctanh t t 13. 2 arccosy x x 14. ( ) arctan(ln )f

Evaluate the integrals in Problems 15-26. Do not use a calculator .

15. 1

4

20

1

1 4dx

x 16.

5

320

2

9 25dx

x 17.

21

0 1

xdx

x 18. 2

8

2 (2 1)dt

t

19. 2

64

wdw

w 20.

216 (ln )

dx

x x 21.

2

arctan

1d

22. 2

5

1

xdx

x

23.

2

2

5

1

xdx

x 24. 2

cos

25 sin

tdt

t 25.

2

43

v

v

edv

e

26. 3

2

3 1

xdx

x

161

x

y

Do not use a calculator on problems 27-32.

27. Find the average value of 2

6

1 (2 )y

x

on the interval

1

20,

.

28. Find the area of the region bounded by 2

3, 0, 0, and 1.5

9y y x x

x

.

29. For 4 cos sin 6 0x y , find 6 4 at ,

dy

dx

.

Integrate in Problems 30-32.

30. 2sec (2 )

tan(2 )

xdx

x 31.

cos

sin

t t

t

e edt

e 32. 2cot (3 )x dx Hint: You need to make a trig

substitution before integrating.

33. Use a calculator to find 1 2g for

3( ) 2 5g x x x .

34. Consider the differential equation given by 2

dy

dx

xy .

a. Sketch a slope field on your own paper for the given

differential equation at the twelve indicated points, and

sketch a particular solution containing the point (0, 1).

b. Use Euler’s Method starting at the point (0, 1) with step size

1x to find an approximate y-value when x = 2.

c. Solve the differential equation 2

dy

dx

xy and find the particular solution containing the point

(0, 1). Give your answer in the form .y f x

d. Use your solution to find the exact y-value when x = 2.

35. The number of rabbits on an island during a measured five year period of time was found to model

the function ( ) 1000sin(2 ) 300cos( ) 3000N t t t on the interval 0,5 years.

a. Use a calculator to find (2) and (4)N N . Specify units and round to the nearest whole

number.

b. Find ( )N t without using a calculator.

c. Use a calculator to find (1) and (3)N N . Specify units and round to the nearest whole

number.

d. At what time(s) during the five years of population measurement was the rabbit population the

greatest? Use an N number “line” (actually, a segment) to justify your answer.

e. What was the greatest number of rabbits on the island during the five years which were

measured?

f. Does ( )N t go through at least one full period (cycle) during the five years?

g. What is the period of ( )N t ?

162

36. In 1990, the population of a city was 123,580. In 2000, the city’s population was 152,918.

Assuming that the population is increasing at a rate proportional to the existing population, use

your calculator to estimate the city’s population in 2025. Express your answer to the nearest

person.

Lesson 6-3 Integration by Parts

Integration by parts is a method of integration used mainly for products of algebraic and transcendental

functions (such as xxe dx ) or products of two transcendental functions (such as sinxe xdx ).

Development of the formula for integration by parts: If u and v are both functions of x, then

d

dxuv

Selected Answers:

1. 3

2. 4

3. 2

3

5. 4x 6. 1x 7. 3

5cos y 8.

3

10sin x

9. 2

6

1 9y

x

11.

2

1

y

y

eg y

e

12. 3

3

2 1

th t

t

13. 2

22 arccos

1

xx x

x

15. 12

16.

30

17. 1

2ln 2

18.

4 2 1

2 2arctan

tC

20.

lnarcsin

4

xC

21. 3

22

3arctan C 23. 5 5arctanx x C 25.

21

2 3 3arctan

veC

26. 5 2

3 32 1

15 33 1 3 1x x C

27.

3

2

28.

2

29. 1

3 30. tan 2x C

31. ln sin te C 32. 1

3cot 3x x C

33. 1.528 or 1.529 34c.

21

4x

y e

35b. 2000cos 2 300sinN t t t d. 3.876 or 3.877 years e. 4217 rabbits g. 2 years

36. 260,456 people

Formula for integration by parts:

uv dx uv vu dx or u dv uv vdu

163

Strategy: Let u be the part whose derivative is “simpler” (or at least no more complicated) than u

itself. Let dv be the more complicated part (or the part which can easily be integrated). Also,

remember that you typically have only two choices. If one choice doesn’t work, try the other.

Example 1. xxe dx Example 2. sin 3x x dx Example 3. arcsin xdx

xxe dx sin 3x x dx arcsin xdx

Example 4. 2 sin 2x x dx Example 5. 2

1ln

e

x x dx Example 6. Complete

the square to find

2

1

4 8dx

x x .

Let u =

du =

Let dv =

v =

164

Assignment 6-3:

Integrate without using a calculator. Integration by parts will be used on most, but not all problems.

1. sinx xdx 2. cos 2x x dx 3. 24 xxe dx 4. 32 xx e dx

5. x

xdx

e 6. ln x dx 7.

2

ln xdx

x 8.

2

ln xdx

x

9. arctan x dx 10. 2 cosx x dx 11. 2 2xx e dx 12. 1

20 1

xdx

x

13. 1

23

1

7 6dx

x x

14.

13

0

xxe dx 15. 2

1ln

e

x x dx 16. arctanx x dx *

Differentiate.

17. 22arctan xy e 18. ( ) ln arcsinf x x

19. ( ) sin(arctan )g t t

Evaluate the expressions. Remember not to use a calculator.

20. 1

2arccos 21. 3

2arcsin

22. 1

3arctan

23. Sketch a triangle to find tan y , given that 2

3arcsiny .

24. 2

6arcsin( 1)x

. Solve for x.

Let R be the region bounded by 2 , 0, 0,

x

y e y x

and , ( 0)x k k .

25. Sketch Region R, set up an integral for its area, and find the area (in terms of k).

26. Find the volume (in terms of k) of the solid formed by revolving Region R about the x-axis.

27. Find the volume (in terms of k) of the solid formed by revolving Region R about the y-axis.

28. If 2 23 2 4 20, find .dy

dxx y xy

29. Find the point(s) at which the graph from Problem 28 has vertical tangents.

30. Find equations of lines tangent and normal to the graph from Problem 28 at the point 20

3, 0 .

31. Find a general solution of 33 0y y x .

32. Find the particular solution of 2 lnxy y x , if the graph of the particular solution contains

the point (e,1). Make sure that your answer expresses y as a function of x. ( y f x )

33. Use Euler’s method with three steps to approximate 0.6y if 0 2y on the solution of

the differential equation y x y . Do not use a calculator.

2

2* can be

1

integrated using

long division

xdx

x

165

Lesson 6-4 Partial Fractions, Mixed Integration

We have often simplified an expression like 1 1

4 3x x

by getting a common denominator and

combining the two fractions into one. By a reverse process we can sometimes split a single fraction in

two to make integration easier.

Example1: Example 2:

2

1

7 12dx

x x 2

5 3

2 3

xdx

x x

Selected Answers:

1. cos sinx x x C 2. 1 1

2 4sin 2 cos 2x x x C

3. 2 22 x x

e ex C

5. x xe ex C 6. lnx x x C 7.

1 1ln x C

x x 8.

31

3ln x C

9. 21

2arctan ln 1x x x C 10. 2 sin 2 cos 2sinx x x x x C

11. 2 2 2 21 1 1

2 2 4

x x xe ex e x C

12.

1

2ln 2 13.

6

14. 32 1

9 9e

15. 43 1

4 4e

16. 21 1 1

2 2 2arctan arctanx x x x C

17.

2

4

4

1

x

x

ey

e

18. 2

1

arcsin 1f x

x x

20.

3

22.

6

23. 2

5 24.

3

2x

25. 22 2k

A e

26. kV e 28. 3 2

2 2

x yy

y x

29. 2, 2 , 2,2

30. tangent: 3 20

2 3y x normal: 2 20

3 3y x 31.

4 4

3 3y x C

32. 21 1

2 2lny x 33. 3.584

166

Example 3: Example 4:

2

2 2

2 3

xdx

x x

3

2

2

2

x xdx

x x

Example 5: Integrate these four “look-alike” integrals. Although they have similar appearances, they

will require you to use three completely different integration formulas.

a. 21

dx

x b.

1

dx

x c.

21

x dx

x d. 2(1 )

dx

x

Example 6: Integrate the following two “look-alikes.”

a. ln

dx

x x b. ln x

dxx

Assignment 6-4

Integrate without using a calculator.

1. 2

1

1dx

x 2. 2

3

2dx

x x 3. 2

5 2

2 1

xdx

x x

4. 2

3

2 2 2x xdx

x x

5. 3 lnx x dx 6. 2 sin 3x x dx 7. 2 3x x

dxx

8. 6(2 1)x dx

9. 2

23 1t dt 10. ln y

dyy 11.

2sec

1 tand

12.

2

2

sec

1 tand

13. 2

2

sec

(1 tan )d

14.

2

2

tan sec

1 tand

15.

22 4

1

xdx

x

16. cos(3 ) sin(3 )ue u du

17. 3

2 3 32sin cosx x x dx 18.

5

2

2x x

x

e edx

e

19.

4

4

1

xdx

x 20.

2

5

16

xdx

x

21. 2

1

10 32dt

t t 22. 2

2 12

4

xdx

x x

23. 0

2 cosx x dx

167

Differentiate:

24.

2 3( )

tan

xf x

x

25.

3( ) ln(sec )h y y 26. cos(arcsin )x t 27. 2arctan( 1)y v

28. For siny x x , evaluate

2

2

d y

dx at

4x

without a calculator.

29. Without using a calculator, find the x-values where sinxy e x has horizontal

tangents on the interval [ , ] .

30. Without using a calculator, find the area of the region bounded by

4

2, 0, and 1

1

xy y x

x

.

31. a. Find a general solution of the differential equation cos( )x

yy

.

b. Write an equation for the particular solution containing the point 2, 3

.

32. Find the slopes of the solution curves for dy

dxxy at the following points:

a. (0,0) b. (0,1) c. (1, 1)

d. (1,1) e. ( 1, 2)

33. a. Sketch the slope field for dy

dxxy , using 2 2x and 2 2y .

Plot your slopes on dot graph paper.

b. Sketch the particular solution which contains the point (0, 2) .

34. Find the particular solution of differential equation dy

dxxy which contains

the point (0, 2) . (Solve for y, and compare the graph of the equation to your sketch from

Problem 33b).

Selected Answers:

1.1 1

2 2ln 1 ln 1 + Cx x 2.

2ln + C

1

x

x

3.

3

2ln 2 1 ln 1 + Cx x

4. 2ln ln 1 ln 1 + Cx x x

5. 2 23 3

2 4lnx x x C

6. 21 2 2

3 9 27cos 3 sin 3 cos 3x x x x x C 7.

3 1

2 24

36x x x C 8.

71

142 1x C

9. 5 39

52t t t C 10.

3

22

3ln y C 12. C 13.

11 tan C

14. ln cos C or 21

2ln 1 tan C 15. 2 2 2ln 1x x x C

168

More Selected Answers:

17. 5

3 22

15sin x C 18. 3 21

3

x x xe e e C 19. 22arcsin x C

20. 21 5

2 4 4ln 16 arctan

xx C 22.

3ln ln 4 + Cx x 23. -4

25. 2

3 ln sec tanh y y y 26. 21

tx

t

27.

4

2 1

1 1

vy

v

28. 2

4 2 24y

29. 3

4 4,x

30.

4

31a. 2sin or 2siny x C y x C b. 2sin 7y x

32a. 0 b. 0 c. 1 d. 1 e. 2 34. 21

22x

y e

Lesson 6-5 Logistic Equations

Exponential growth modeled by kty Ce assumes unlimited growth and is unrealistic for most

population growth. More typically the growth rate decreases as the population grows and there is a

maximum population M called the carrying capacity. This is modeled by the logistic differential

equation dP

dtkP M P .

The solution equation is of the form 1 kMt

MP

Ce

. Note: Unlike in the exponential growth

equation, C is not the initial amount.

Example: A national park is capable of supporting no more than 100 grizzly bears. We model the

equation with a logistic differential equation with 0.001k .

a. Write the differential equation.

b. The slope field for this differential equation is shown.

Where does there appear to be a horizontal asymptote?

What happens if the starting point is above this asymptote?

What happens if the starting point is below this asymptote?

c. If the park begins with ten bears, sketch a graph of P t on the slope field.

t

P

t

169

d. Solve the differential equation to find P t with this initial condition.

e. Instead of solving the differential equation, use the general form f. Find limt

P t

of a logistic equation 1 kMt

MP

Ce

to find the same solution.

g. When will the bear population reach 50? h. When is the bear population growing the fastest.

Assignment 6-5

1. Logistic growth has 0.00025k , 200M , and 0 10P .

a. Write a differential equation for the population.

b. Sketch the population function on the slope field.

c. Find a formula for P.

Show all steps using partial fractions.

d. When is the population growing the fastest?

t

P

170

2. If 2.04 .0004dP

dtP P find k and the carrying capacity.

Match each of these differential equations with one of slope fields shown.

3. .065dy

dxy 4. .0006 100

dy

dxy y 5. 150

.06 1dy y

dxy

A. B. C.

6. Given the logistic equation 0.6

2000:

1 19 tP t

e

a. find the carrying capacity. b. find the value of k.

c. find the initial population. d. find the time at which the population reaches 500.

e. give the logistic differential equation.

7. Given the logistic differential equation .03 100 :dP

dtP P

a. find the value of k. b find the carrying capacity.

c. find the value of P when dP

dt is the greatest.

8. Given the logistic differential equation 2 1 :50

dy

dt

yy

a. find the value of k. b. find the value of M.

c. give the logistic equation if y (0) = 10.

9. A 200 gallon tank can support no more than 150 guppies. Six guppies are introduced into the

tank. Assume that the rate of growth of the population is .0015 150dP

dtP P , where time t

is measured in weeks.

a. Find a formula for the guppy population in terms of t.

b. How long will it take for the guppy population to be 100? 125?

10. The amount of food placed daily into a biology lab enclosure can support no more than 200

fruit flies. A biologist releases 25 flies into the enclosure. Four days later she counts 94 flies.

a. Give the logistic equation. b. Find the number of flies on the 7th day.

c. When will there be 175 flies? d. Find the logistic differential equation.

e. Find the population and the time at which the growth is the fastest.

f. Starting with 94 flies on day 4 and a step size of 1 day, use Euler’s Method to

x

y

x

y

x

y

171

approximate the number of flies on the 7th day.

Evaluate these integrals without a calculator.

11. 23

0sec d

12.

1

30

36

2 1

dx

x 13.

1

xdx

x (hint: let )u x 14. tan 24

0secxe x dx

15. cos

2 sin

xdx

x 16. 2 5

tdt

t 17. tan ln y

dyy 18.

ln

dx

x x

19. 2 cosx xdx 20. 1

2sin x dx

Find the indicated limits without a calculator.

21. 2

2lim

2x

x

x

22.

2

2lim

2x

x

x

23.

2

1

2lim

1x

x x

x

24.

1

2 1lim

1x

x

x

25. For

2

3

3, 1

4, 1( )

2 , 1 2

8, 2

x x

xf x

x x

x x

1 2

Find a. lim ( ) ? b. lim ( ) ?x x

f x f x

26. t x

t

d

dtxe dx

= ? 27. 2

3(1 sin )

xd

dxt dt = ? 28. 4

3

2?

d

dx

xx

x

29. Find a and b so that ( )f t is differentiable at 1t .

3 2

2

2, 1( )

4, 1

at bt tf t

bt at t

30. 3 3( )y t t t represents the position of a point on the y-axis at time 0t .

( )v t represents the velocity and ( )a t represents the acceleration of the point.

Without a calculator, find

a. (2)y b. (2)v c. (2)a

Find the indicated derivatives.

31. 2cos lny t 32. ( ) tan( ) ln 1f x x x 33. 2

2

sin?

d

dt

t

t

?y

( ) ?f x

172

Lesson 6-6 L’Hospital’s Rule

Some limits cannot be found using algebraic methods. If direct substitution produces one of these two

indeterminate forms 0

or 0

, then a rule known as L’Hospital’s Rule may help you find the limit.

.

L’ Hospital’s Rule

If ( ) ( )

lim ( ) 0 and lim ( ) 0 or if both of these limits are , then lim lim( ) ( )x c x c x c x c

f x f xf x g x

g x g x→ → → →

= = =

.

To use L’Hospital’s Rule, you must have the limit of an expression which is written in fractional form.

Examples: Evaluate.

1. limxx

x

e→ 2. lim

x

x

e

x→ 3.

0lim

xx

x

e→

SELECTED ANSWERS

1a. ( ).00025 200dP

dtP P= −

c. ( ) .05

200

1 19 tP t

e−=

+ 2. .0004k = , 100M =

6a. 2000 b. .0003 c. 100 d. 3.076 e. ( ).0003 2000dP

dtP P= −

8c. 2

50

1 4 ty

e−=

+

9a. ( ) .225

150

1 24 tP t

e−=

+ 10a. ( ) .456

200

1 7 tP t

e−=

+ b. 155 flies c. 8.526 days

10d. ( ).002 200dP

dtP P= − 11. 3 12. 8 13.

12 ln

1

xx C

x

−+ +

+ 14. 1e−

16. ( )21

2ln 5t C+ + 18. ln ln x C+ 19.

2 sin 2 cos 2sinx x x x x C+ − +

20. cos sinx x x C− + + 21. -1 22. DNE 23. 3 25a. -2 26. t tte te−−

28. 3

3 441

42 6x x x

−− −+ − 29. ,2 1a b= − = − 30b.

3

1612 c. 5 5

812 12 2 11−

•− =

31. ( )22sin ln t

yt

− = 33.

2 2

4

2sin cos sin 2t t t t t

t

• • •−

173

4. 0

limx

x

e

x 5.

3

0

3 3lim

x

x

e

x

6. 2lim x

xx e

7. 1

1 1

ln 1limx x x

8.

2

1

2 2lim

1x

x

x

(w/o LR) 9.

2

1

2 2lim

1x

x

x

The following limits are not 0

or 0

forms. Identify the form and tell which are indeterminate.

10. 1lim 1

x

x x 11.

0lim x

xx

12.

1

lim x

xx

13.

1

0lim sin x

xx

Find the following limits.

14. 1lim 1

x

x x

15.

1

lim x

xx

174

16. Sometimes limits can be evaluated quickly using the concept of relative growth rates. Rank the

following in order of rate of growth as x approaches infinity from slowest to fastest.

2 10, 1, , ln , ,= = = = = = xy x y y x y x y x y e

Use the concept of relative growth rates to evaluate the following when possible.

17. 3

lnlim

xx

x

e x→ + 18.

0lim

ln

x

x

e

x+→

Assignment 6-6

Find the indicated limits without using a calculator.

Hint: Not all problems will require the use of L’Hospital’s Rule.

1.

2

2

4lim

2x

x

x→

−

+ 2.

2

1

2 3lim

1x

x x

x→

+ −

− 3.

20

1lim

x

x

e x

x→

− − 4.

0lim

(1 )xx

x

x e→ − −

5. 0

limx

x

x+→ 6.

0limx

x

x−→ 7.

0limx

x

x→ 8.

2

2

4lim

ln( 1)x

x

x→

−

−

9. 20

2( 1)lim

x

x

e

x→

− 10.

5 2

5 4

2lim

3 5x

x x

x x x→

−

+ − 11.

4 2

5 4

2lim

3 5x

x x

x x x→

−

+ − 12.

2lim

x

x

e

x→

13.

5 4

4 2

3 5lim

2x

x x x

x x→

+ −

− 14.

2 5limx

x

x→

+ 15.

2 5lim

x

x

x→−

+ 16.

5

2 10lim

5x

x

x→

−

17. 0

tanlim

secx

x

x x→ 18.

4

cos sinlim

2 2 tant

t t

t→

−

− 19.

2

sin(2 )lim

cos

→

20. 1

4arctan( )

lim1x

x

x

→

−

−

21.

( )3

0lim x x

xe x

+→+ 22. ( )

1

0lim 1 x

xx

+→+ 23.

( )

1

lim 2 x

xx

→+ 24. ( )

1

1lim ln

x

xx

+

−

→

25. Use the concept of relative growth rate to evaluate the following.

a. 2

limx

x

e x

x→

− b.

lnlimx

x

x→

26. Use a calculator to find

3 2

3 21.5

2 3 8 12lim

6 25 34 15x

x x x

x x x→

− − +

− + −

175

Antidifferentiate

27. 2 5( ) xf x e x+ = 28. 2

2 1( )

t t

tx t

e −

− =

Evaluate without using a calculator.

29. 3

2

lne xdx

x 30. 1

03 x dx−

31. 2

2

4 5

tdt

t t

−

− − 32. 2

1

4 12dt

t t− + +

33. What interest rate would be required for a savings account to triple in value every 20 years, if

interest is continuously compounded? ( )rtA Pe= Use your calculator, and express your answer

to the nearest hundredth of a percent.

34. A population of rabbits is given by the formula ( ) 4.8 .7

1000

1 tP t

e −=

+ where time is measured in months.

Find: a. the value of k. b. the carrying capacity.

c. the initial number of rabbits. d. the time at which the population is growing the fastest.

35. A wild animal preserve can support no more than 250 lowland gorillas. Twenty eight gorillas

were known to be in the reserve in 1970. Assume that the rate of growth of the population is

( ).0004 250dP

dtP P= − where t is measured in years. Find a formula for the gorilla population.

When will the population reach 249 gorillas?

Use the differential equation 2y y xy − = for Problems 36-41.

36. Solve for y and find slopes for the solution curve(s) to the differential equation at:

a. (0,2) b. ( 2,0)− c. ( 1,1)−

d. ( 4,1)− e. ( ,0)k where k is a constant

37. Sketch a slope field on your own paper for the equation

(in Quadrant 2 of a coordinate plane - like the one shown

at the right).

38. Sketch a solution curve containing the point (0, 2).

39. Use Euler’s method with the same starting point and 1

3x = − to approximate ( )1y − .

40. Solve the differential equation, and find the particular solution containing the point (0, 2). Use

your calculator to graph the particular solution, and compare it to your graph from Problem 38.

41. Find the exact value of ( )1y − .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . x

y

1

1−

176

42. Let R be the region enclosed by the graphs of ( )3cosy x= and 2y x= . Set up and evaluate

an integral that gives the area of R. (Show the equation used to find the limits of integration.

Make sure that your calculator is in radian mode.)

43. A particle moves along the x-axis so that, at any time 0t , its velocity is given by

( ) ln( 1) 2 1v t t t= + − + . Find the total distance traveled by the particle from 0 to 2t t= = .

(Show an integral set up and an answer.)

44. Integrate 2

3 3.

9

xdx

x

−

−

Selected Answers:

1. 0 2. 4 3. 1

2 4.

1

2 6. 1− 8. 4 10. 2

3 11. 0 12. or DNE

14. 1 16. 0 17. 1 18. 1

2 2 19. 2 20.

1

2 21. 6e 22. e 23. 1

24. 1 27. ( )2 5

2

xef x C

+

= + 28. ( ) 2

1t t

x t Ce −

= − + 29. ( )4

33 3

4 4ln 2−

30. 1 1

3ln3 ln3− + 31. 21

2ln 4 5t t C− − + 32.

2arcsin

4

tC

−+ 33. 5.49%

34c. 8 rabbits d. 6.857 months 35. ( ) .1

250

1 7.929 tP t

e−=

+, about year 2046

36a. 4 c. 1 e. 0 40. 21

222

x x

y e+

= 42. 1.248 or 1.249 43. 1.540

44. 2ln 3 ln 3x x C+ + − +

177

CALCULUS EXTENDED UNIT 6 SUMMARY

Euler’s Method: This is a method of finding an approximate y-value on a solution to a differential

equation.

Definition of the Inverse Trig Functions:

Function Domain ( x values) *Range ( y values)

arcsin siny x y x= = 1,1−

−

arccos cosy x y x= = [ 1,1]− [ , ]0

arctan tany x y x= = ( , )−

−

*A geometric representation of the range values

for each inverse trig function is shown in the

coordinate plane at right.

( )slopedy

dxx y y x =

0

2

2

−

arccos

arcsin

arctan

178

Derivatives of the Inverse Trig Functions:

2

1arcsin

1

d

dxx

x=

−

2arcsin

1

d

dx

uu

u

=

−

2

1arccos

1

d

dxx

x

−=

−

2arccos

1

d

dx

uu

u

− =

−

2

1arctan

1

d

dxx

x=

+

2arctan

1

d

dx

uu

u

=

+

(where u is a function of x)

Integration of Inverse Trig Functions

1. 2 2

arcsinu u

dx Caa u

= +

− 2.

2 2

1arctan

u udx C

a u a a

= +

+

Partial Fractions: Example: Write fractions like ( )( )

2

3 4x x− − as

( ) ( )3 4

A B

x x+

− − then find A

and B in order to integrate.

Complete the Square: When the denominator of an integrand fraction is not factorable completing the

square may allow you to use an inverse trig formula.

Logistic Equations: (M is the carrying capacity) ( )dP

dtkP M P= −

1 kMt

MP

Ce−=

+

L’Hospital’s Rule

If ( ) ( )

lim ( ) 0 and lim ( ) 0 or if both of these limits are , then lim lim( ) ( )x c x c x c x c

f x f xf x g x

g x g x→ → → →

= = =

To use L’Hospital’s Rule, you must have the limit of an expression which is written in fractional form.

For limits in other indeterminate forms such as 0 01 , , or 0 use logarithms to rewrite them in fraction

form.

Formula for Integration by Parts:

u dv uv v du= −