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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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
10. Consider the differential equation
and let y = f(x) be a solution.
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
Calculus Review - Calculator
10. Consider the differential equation
and let y = f(x) be a solution.
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
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
Calculus Review - Calculator
14. A function f is defined on the interval [0,4] x [-2,5], and its derivative is
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
Calculus Review - Calculator
14. A function f is defined on the interval [0,4] x [-2,5], and its derivative is
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
sin' ( ) 2cos3xf x e x
Calculus Review - Calculator
14. A function f is defined on the interval [0,4] x [-2,5], and its derivative is
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
sin' ( ) 2cos3xf x e x
Calculus Review - Calculator
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
Calculus Review - Calculator
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.
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
Calculus Review - Calculator
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.
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
Calculus Review - Calculator
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.
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