calculus review - calculator 1. let h(x) be the anti- derivative of g(x). if - 1

Calculus Review - Calculator

1. Let h(x) be the anti-derivative of g(x). If

-

1

3 11( ) 1 and ln ,

4 12

fi nd h(-2)

x xg x e e h

12

3 2

3 2

3 2

3 23

ln4

3 2

3 2

( ) (1 ) 1 ,

232

13

We know that...

3 11(ln ) need to fi nd C

4 122

( ) 13

11 21

12 3

11 2 31

12 3 4

11 2 112 3 4

11 212

x x x x

x

x

h x e e dx u e du e

u du

u C

e C

h

h x e C

e C

C

C

3 2

3 22

13 8

11 112 121

2So, ( ) 1 1

3Using you calculator fi nd h(-2)

2So, ( 2) 1 1 .464

3

x

C

C

C

h x e

h e


2. The acceleration of a particle moving on a line is given by:

If the particle starts from rest, what is the distance traveled from t=0 to t=3.61?

-

2

1( ) 3a t t

t

12 12

12 3 2

12 3 2

12 3 2

12 3 2

3 2 5 2

3 2 5 2

( ) ( 3 )

2 2

Need to fi nd C when t=0

(0) 0

0 2 0 2 0

0

Velocity f unction is: v(t) = 2 2

Need to fi nd s(t)

( ) 2 2

2 2 2 21 3 1 5

4 4( )

3 5Find t, when

v t t t dt

t t C

v

C

C

t t

s t t t dt

t t C

s t t t C

3 2 5 2

3 2

3 2 5 2

s(t)=0

4 40

3 54 4

0 so 0 and 13 5

1 is not in the interval [0,3.61]

Using the calculator, fi nd s(3.61) and s(0)

4 4(3.61) (3.61) (3.61)

3 528.954

(0) 0

0

Distance travele

t t

t t t t

t

s C

C

s C

C

d is 28.954


3. On each point (x, y) on a curve, the slope of the curve is

If the curve contains the point (0,7), then which of the following is the equation of the curve?

-

3

3

3

3

3

3

2

2

3

3

ln 6

0

3 ( 6)

3( 6)

3ln 6

3ln 6

6

6

when x = 0 y = 7

7 6

1

6 Answer: E

y x C

x C

x

x

dyx y

dxdy

x dxy

xy C

y x C

e e

y e e

y Ce

Ce

C

y e

23 ( 6)x y

3

3

3

3

2 3

A. 6 1

B. 7

C. 7

D. 49

E. 6

x

x

x

y e

y x

y e

y x

y e


4. The volume, V (cubic inches) of unmelted ice remaining from a melting ice cube after t seconds is given by

How fast is the volume changing when t = 40 sec?

-

4

2( ) 2000 40 0.2V t t t

2

3

( ) 2000 40 0.2

40 .4

at 40

40 .4(40)

40 16

24 in / sec

V t t t

dVt

dtt

dVdt


5.

- T1 T22

-2 -1 1

5

1

21x dx

Area T1 + Area T2

1 1(1)(1) (2)(2)

2 21 5

22 2


6. If the function f is defined as

on the interval[-7,5], then g(x) has a local minimum at x = ?

-

-2 0 5 x

+ -1 - 4 + g(x)

inc dec inc

There is a local min at x = 4

6

2

0( ) ( 3 4)

xg x t t dt

2

0

2

2

2

Find f (x)

( 3 4)

3 4

'( ) 3 4

0 3 4

0 ( 4)( 1)

4 or 1

xdt t dt

dxx x

g x x x

x x

x x

x x


7. The normal line to the graph

at x = 1 also intersects the graph at which one of the following values of x?

A. -1.18

B. -1.11

C. -1.06

D. -0.98

E. -0.86

-

7

4 1y x

4 4

3 3

4

( ) 1 (1) 1 1 2

'( ) 4 ' (1) 4(1) 4

1Normal Slope =

4When 1, 2

1Equation: 2 ( 1)

4Find the points of intersection

11 ( 1) 2

4Use the calculator to fi nd x

1.11 1

1.11

f x x f

f x x f

x y

y x

x x

x x

x

Ans: B


8. Let R be the region in the first quadrant bounded above by y = 4x + 3 and below by

Find the area of R

-

Bottom

Find the point of intersection

8

2 3y x

Top

2

2

4 2

0

4 3 3

0 4

0 ( 4)

4 0

Area

4 3 ( 3)

Use nI nt(4x+3-(x 2̂ 3), ,0,4) 10.667

x x

x x

x x

x x

x x dx

x


9. A conical cup 8 inches across the top and 12 inches deep is leaking water at the rate of 2cu in /min

A. At what rate is the water level dropping when the water is 6 in deep.

-

9

2

2 3

2

2

2

13

4 12

3

1fi nd V=

3 3 27

327

9

2 when h = 6

-2 6936

29

2 11 41

or -.159 in/ min2

V r h

r h

hr

dh h hh

dt

dV dhh

dt dt

dhh

dtdVdt

dhdtdhdtdhdt

dhdt


9. A conical cup 8 inches across the top and 12 inches deep is leaking water at the rate of 2cu in /min

B. At what height is the water level dropping when the cup is half full?

-

10

23

3

3

3

3

3

3

1Full Volume = 4 (12)

364

Half Volume 32

3227

2732

32 27

864

6 4 or 9.524

in

in

h

h

h

h

h h


10. Consider the differential equation

and let y = f(x) be a solution.

A. On the axis provided, sketch a slope field at the 20 indicated points.

-

11

12

dy y

dx

(x,y) 0 1 2 3

0 1 1 1 1

1 ½ ½ ½ ½

2 0 0 0 0

3 - ½ - ½ - ½ -½

4 -1 -1 -1 -1




B. Find the general solution y = f(x)

-

12

12

dy y

dx

1ln2 2

12

12

12

21

2 22

21

2 2

1 ln2

21

ln22

2

2

2

x Cy

x

x

x

dy y y

dxy

dy dx

dydx

y

y x C

y x C

e e

y Ce

y Ce

y Ce




C. Find the particular solution to the differential equation with the initial condition

f(2 ln 3) = 4

-

13

12

dy y

dx

1

12

1(2ln3)

2

(ln3)

(ln3)

1ln

3

12

When x = 2 ln 3, y = 4

2

4 2

4 2

4 2

2

12

3

6

6

2 6

x

x

y Ce

Ce

Ce

Ce

Ce

C

C

C

y e


11.Find the approximate volume generated by revolving the first quadrant area enclosed by y = 3x + 4,

and the y-axis about the x-axis

- Outer

inner

Find the points of intersection using your calculator:

(.966, 6.90)

Use the washer method

14

2xy e

.966 2 2

0.966 2 2 2

0

( )

((3 4) ( ) 55.59x

outer inner dx

x e dx


12. A ball is thrown from the top of a 1200-ft building. The position function expressing the height h of the ball above the ground at any

time t is given as

Find the average velocity for the first 6 seconds of travel

-

15

2( ) 16 10 1200h t t t

2

2

(6) (0)Find

6 0(6) 16(6) 10(6) 1200 564

(0) 16(0) 10(0) 1200 1200

564 1200 636106f t/ sec

6 0 6

h h

h

h


13. For the ellipse

what is the value of

at the point in the

third quadrant where x = -1

-

16

2 22 11x y 2

2

d y

dx

2 2

2 2

2

2

2 2

2

2 2

2 11

2( 1) 11

2 11

9

3

Function is in the third Quarter so,

-1 = - 3

2 11

4 2 ' 0

4 22

When x = -1, y = -3

2( 1) 23 3

( 3)( 2) [ 2( 1)( 2) 2 '

x y

y

y

y

y

x and y

dx y

dxx yy

dy x xdx y y

dy

dx

y xyd y

dx y

2

23

( 3)

46

3 18 4 1 229 3 9 27


14. A function f is defined on the interval [0,4] x [-2,5], and its derivative is

A. Sketch f’ in the window

[0,4] x[-2,5]

Use your calculator to find the graph then do a rough sketch

17

sin' ( ) 2cos3xf x e x



B. On what interval is f increasing? Justify your answer.

f is increasing when f’(x) >0, this is true for all values between a and b

Find where

Using calculator

A = 0.293 < x < 3.760

18

sin' ( ) 2cos3xf x e x 2cos(3 ) 0xe x



C. At what value(s) of x does f have local maxima? Justify your answer.

-

- + -

0 a b 4

Since f decreases to the right of endpoint x = 0 f has a local maximum at x = 0. There is a local max at x = 3.760 because it changes from increase to decrease

19




D. How many points of inflection does the graph of f have? Justify your answer.

-

0 P Q R 4

inc dec inc dec f’(x)

+ - + - f”(x)

cu cd cu cd f(x)

Since the graph of f changes concavity at p, q, and r there are 3 points of inflection

20



15. The rate of sales of a new software product is given by

where S is measured in thousands of units sold per month and t is measured in months from the initial release of the product on January 1, 2007.

A. This product initially sold at the rate of 2500 units per month, and the sales rate has doubled every three months. Find C and k

-

21

( ) ktS t Ce

0

3

3

3

When t = 0, S(t) = 2.5

( )

2.5

2.5

I f the rate doubles every 3 months

then S(t) = 5 when t = 3

5 2.5

2

ln2 ln

ln2 3

ln20.221

3

kt

k

k

k

k

S t Ce

Ce

C

e

e

e

k

k




B. Find the average rate of sales for the first year.

-

22

( ) ktS t Ce 12

0

12 0.25

0

1Avg = ( )

121

2.51213.525 thousands or 13525 units/ month

S t dt

e dt




C. Using the midpoint rule with three equal subdivisions, write an expression that approximates

3 equal subdivisions at 4, 5, 6, 7

With delta x = 1, midpoints will be at

4.5, 5.5, and 6.5

Midpoint Rule

23

( ) ktS t Ce

7

4( )S t dt

7

40.231(4.5) 0.231(5.5) 0.231(6.5)

( ) ( (4.5) (5.5) (6.5))

1(2.5 2.5 2.5 )

S t dt x S S S

e e e




D. Using correct units, explain the meaning of

in terms of software sales.

-Represents the number of units

sold, in thousands, during May, June, July in 2007

24

( ) ktS t Ce

7

4( )S t dt

7

4( )S t dt

calculus review - calculator 1. let h(x) be the anti- derivative of g(x). if - 1

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