NEED HELP WITH CALCULUS REVIEW

Math 207 Final Review page 1

1. The picture shows the graph of a function f. Sketch the graph of each of the following. a) g (x) = f 0 (x) b) F (x) where F 0 (x) = f (x)

2. Prove that d

dx

� sin�1 x

� =

1 p 1�x2

.

3. State the Intermediate Value Theorem.

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y

x

y

4. Find each of the following limits.

a) lim x!5

x2 �25 x2 �2x�15

b) lim x!�3�

x2 �25 x2 �2x�15

c) lim x!�3

x2 �25 x2 �2x�15

d) lim x!1

x2 �1 2x +1

e) lim x!1

� log2

� 4x2 �1

� � log2

� x2 +1

��

f) lim x!3+

2(x+1)(x�3)(x+5) (x�1)(x�3)

g) lim x!1

lnx

x

h) lim x!0+

1�ex

(x�1)2 �1

i) lim x!4+

p x�2 4�x

j) lim x!0

sinx�x x3

k) lim x!0�

1� cos2 x sin3 x

l) lim x!1

� 1�

2

x

�x m) lim

x!1 tan�1 x

5. Di¤erentiate each of the following.

a) f(x) = ln 3 p x5 �2x+4

b) f(x) = 2 x+1 x�1

c) f(x) = �3e�4x2

d) f (x) = x � ln5+lnx2

� �2x

e) f (x) = ln � sin2 (�x�1)+1

� f) 10 = x2y �xy2 +y3 �x3

g) f (x) = esinx +ecosx

h) f (x) = 3x +x3

i) y = tan�1 � x4 �

j) f (�) = cos2 (2�)+sin2 (2�)

k) f (x) = esinx

ecosx

l) g (x) = ln(secx+tanx)

m) f (x) = � sin�1 x

�4 +x

n) f (x) = 1

3 xe3x �

1

9 e3x

o) f (�) = sin5�cos5�

6. Di¤erentiate each of the following.

a) d

dx

0 @ xZ

0

� 6t2 �4t+1

� dt

1 A

b) d

dx

0 B@x

2Z 0

� 6t2 �4t+1

� dt

1 CA

c) d

dx

0 @sinxZ

0

� 6t2 �4t+1

� dt

1 A

d) d

dy

0 @ yZ

0

p ln4 x+10dx

1 A

e) d

dt

0 B@ p tZ 0

� 1

1+x2

� dx

1 CA

f) d

dx

0 @10Z x2

y2

1+y3 dy

1 A

g) d

dy

0 B@ eZ lny

ex

1+ex dx

1 CA

h) d

dx

0 @ 1Z x4

ye�y 2 dy

1 A

7. Compute y00 if x3 +y3 = 2.

8. Find all values of c that satisfy the conclusion of the Mean Value Theorem for each of the following.

a) f (x) = x3 �x2 +5 on [1;4] b) g (x) = 1

x +x on [1;5]

c Hidegkuti, 2014 Last revised: November 27, 2014

Math 207 Final Review page 2

9. a) Compute the left and right Riemann Sums for

1Z 0

e�x 2 dx using a uniform partition with n = 5.

b) Compute the exact values for the left and right Riemann Sums for

2Z 0

x3dx using a uniform partition with

n = 100.

10. Compute each of the following integrals.

a)

1Z 0

6x2 �4x+5 x+1

dx

b) Z

1p 1�9y2

dy

c) Z 5xdx

d) Z 3x�1 x+7

dx

e) Z sin2 �d�

f)

5Z 0

e�xdx

g) Z

e�x

e�x +1 dx

h) Z

e�x

(e�x)2 +1 dx

i)

5Z 1

p 2x�1dx

j)

�=2Z 0

sinxcosx dx

k)

�=4Z 0

tanx dx

l)

28Z 2

(x�1)2=3 dx

m) Z xe�x

2 dx

n)

5Z 0

xe�x 2 dx

o)

1Z 0

xe�x 2 dx

p)

�=3Z 0

sin5xcos5xdx

q) Z

1

2+x2 dx

r) Z

1

x p x�1

dx

11. Let f be a function given by f (x) = ln �

x

x�1

� a) Find the domain of f. c) Find the formula for the inverse of f.

b) Find the value of the derivative of f at x = �1.

12. We are on the surface of the Moon. The gravitational acceleration there is g = �1:6 m

s2 . A rock is thrown

vertically upward, from an initial height of 19:2m, with an initial velocity of 8 m

s .

a) Find the velocity function v (t) of the object. b) Find the location function s(t) of the object. c) Find the maximal height that the rock will reach.

d) How long until the rock hits the ground? e) What is the velocity of the rock when it hits the ground?

13. A particle starts at t = 0 and moves along the x�axis so that its position at any time t � 0 is given by x(t) = (t�1)3 (2t�3). a) Find the velocity function v (t) of the particle. b) For what values of t is the object moving to the left? c) Find all values of t for which the object is moving and but its acceleration is zero.

14. A function f has derivative f 0 (x) = �18(x+5)3 (x+4)2 (x+2)x6 (2�x)5 (4�x)2. a) Plot the graph of f 0. b) Find all critical points of f and classify each of them as a maximum, minimum, or a point of in�ection. c) How many points of in�ection does f have?

c Hidegkuti, 2014 Last revised: November 27, 2014

Math 207 Final Review page 3

15. Write the equation of the degree 3 polynomial P (x), given that

a) P (0) = �2, P 0 (0) = 3, P 00 (0) = 12, and P 000 (0) = �240. b) P has a relative minimum at x = �3 and a relative maximum at x = 1; P (0) = 0; and P (1) = 5.

16. Find an equation for all tangent lines drawn to the graph of 2x2 +y2 = 5y �x at x = �2.

17. A company estimates that the total cost of producing q units is C (q) = q3 �155q2 +6375q +3000 a) What is the �xed cost? b) At what level of production will the total cost be minimized? What is the minimal cost? c) At what level of production will the pro�t be maximized, provided that we can sell every item we produce, for $775?

18. Three sides of a symmetric trapezoid are 1 unit long as shown on the picture. Find the fourth side so that the area of the trapezoid is the greatest.

19. Let f (x) = 2x7 � 7x4 + 70x + 4. Let g (x) be the inverse of f (x). Find g0 (4).

20. A town is of a circular shape. The area of the town is growing with a constant 3� mi2

y (square mile per

year). How fast is its radius changing when the radius is exactly 5 miles long?

21. A tank, shaped like a cone held with its circular base upward, is being �lled up with water. The top of the

tank is a circle with radius 5 ft, its height is 15ft. Water is added to the tank at the rate of V 0 (t) = 2� ft3

min .

How fast is the water level rising when the water level is 6ft high?

22. A virus is spreading through a population in a manner that can be modeled by the function g (t) = A

1+Be�t where A is the total population, g (t) is the number infected at time t; and B is a constant. What proportion of the population is infected when the virus is spreading the fastest?

23. A company has $120000 to spend on the development and promotion of a new product. The company

estimates that if x is spent on the development and y is spent on promotion, then approximately x1=2y3=2

400000 items of new product will be sold. Based on this estimate, what is the maximum number of products that the company can sell?

24. A company can sell 20 products if it charges $40 per product. For each dollar decrease or increase in the price, the company can sell one more or one less product, respectively. The total cost of producing q products is C (q) = 32q + 100: What is the maximum pro�t that the company can achieve from manufacturing and selling this product?

25. Find the volume of the right circular cone of the greatest volume that can be written into a sphere with radius R.

26. The derivative of a particle�s horizontal position x(t) is dx

dt = �8, the derivative of its vertical position y (t)

is dy

dt = 1. How fast is the distance of the particle from the origin changing when the particle is at the point

(3;4)?

27. A kite 100 meters above the ground moves horizontally at a speed of 8 m

s . At what rate is the angle between

the string and the horizontal decreasing when 200 meters of string has been let out?

c Hidegkuti, 2014 Last revised: November 27, 2014

Math 207 Final Review page 4

28. Let f be the function that is given by f (x) = ax+ b

x2 � c and that has the following properties.

i) The graph of f is symmetric with respect to the y�axis. ii) lim x!2+

f (x) = 1 iii) f 0 (1) = �2

a) Determine the values of a, b; and c. b) Write an equation for each vertical and each horizontal asymptote of the graph of f. c) Sketch the graph of f (x).

Answers

1. a) g (x) = f 0 (x) b) F (x) where F 0 (x) = f (x)

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y

x

y

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x

y

x

y

2. see handout 3. see handout

4. a) 5

4 b) �1 c) unde�ned d) 0 e) 2 f) 32 g) 0 h)

1

2 i) �

1

4 j) �

1

6 k) �1

l) 1

e2 m)

�

2

5. a) 5x4 �2

3(x5 �2x+4) b) ln2 �2

x+1 x�1 �

�2 (x�1)2

c) 24xe�4x 2

d) 2lnx+ln5

e) 2� sin(�x�1)cos(�x�1)

sin2 (�x�1)+1 f) y0 =

2xy �3x2 �y2 2xy �x2 �3y2

g) cosxesinx � sinxecosx h) 3x2 + (ln3)3x

i) y0 = 4x3

x8 +1 j) f 0 (�) = 0 k) f 0 (x) = (cosx+sinx)esinx�cosx l) g0 (x) = secx

m) f 0 (x) = 4 � sin�1 x

�3 p 1�x2

+1 n) f 0 (x) = xe3x o) 5cos10x

6. a) 6x2 �4x+1 b) 12x5 �8x3 +2x c) � 6sin2 x�4sinx+1

� cosx d)

p ln4 y +10

e) 1

2 p t(1+ t)

f) �2x5 x6 +1

g) � 1

y +1 h) �4x7e�x8

7. y00 = � 2x

y2 � 2x4

y5 = �

2x4 +2xy3

y5

8. a) 8

3 b)

p 5

9. a) left: 0:8075804 right: 0:6811563 b) left: 3:9204 right: 4:0804

c Hidegkuti, 2014 Last revised: November 27, 2014

Math 207 Final Review page 5

10. a) 15ln2�7 b) 1

3 sin�1 (3y)+C c)

5x

ln5 +C d) 3x�22ln jx+7j+C e)

1

2 � �

1

4 sin2� +C

f) 1� 1

e5 g) � ln(e�x +1)+C h) tan�1 (ex)+C i)

26

3 j) 1

2 k)

ln2

2 l) 726

5

m) � 1

2 e�x

2 +C n)

1

2 �

1

2e25 o)

1

2 p)

3

40 q)

p 2

2 tan�1

p 2

2 x

! +C r) 2tan�1

�p x�1

� +C

11. a) (�1;0)[ (1;1) b) � 1

2 c) f�1 (x) =

ex

ex �1

12. a) v (t) = �1:6t+8 = �1:6(t�5) b) s(t) = �0:8t2 +8:0t+19:2 = �0:8(t+2)(t�12)

c) 39:2m d) 12s e) �11:2 m

s

13. a) v (t) = (t�1)2 (8t�11) b) [0;1)[ � 1; 11

8

� c)

5

4

14. a)

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x

y

x

y

b) f has a minimum at x = �5; a point of in�ection at x = �4; a maximum at x = �2; a point of in�ection at x = 0; a minimum at x = 2; and a point of in�ection at x = 4 c) 8

15. a) P (x) = �40x3 +6x2 +3x�2 b) P (x) = �x3 �3x2 +9x

16. y = �7x�12 and y = 7x+17

17. a) $3000 b) $31125 when q = 75 c) q = 80

18. 2 19. 1

70 20. 0:3

mi

y 21.

1

2

ft

min 22.

1

2 23. 11691

24. $96 25. 32

81 �R3 26. �4 27. �0:02

rad s

28. a) a = 0; b = 9; c = 4 b) vertical: x = 2; x = �2, horizontal: y = 0

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1 0

5

0

- 5

- 1 0

x

y

x

y

c Hidegkuti, 2014 Last revised: November 27, 2014