1
MATH 131 Exam 2
, f   x  is negative 
Monday, October 19th
2.6, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4, 3.7, 3.8, 3.9
 Antiderivatives of f is a function F , such that
• 2.6 – Derivatives and Rates of Change
 Tangent to a Curve y  f  x  at x  a is
y  f a  m x  a

F  f
• 3.1 – Derivatives




y  x n , y  n  x n 1
y  c , a constant, y  0
 Power rule:

 Constant multiple rule: y  ax n , then
y  m  x  m  a  f a

f is concave down
y   a  n  x n 1
f a  h  f a
 f a
h 0
h
f a  h  f a
s
average velocity 

t
h
f a  h  f a
velocity v  a   lim
h 0
h
f a  h  f a
derivative f   a   lim
h 0
h
y f  x2   f  x1 
rates of change:

x
x2  x1
m  lim
 marginal cost: C   x 
 Sum rule: h  x   f  x   g  x  ,
h  x   f   x   g   x 
 Difference rule:
h  x   f   x   g   x 

y  e x , y  e x

y  a x , y  a x ln a
• 3.2 – Product and Quotient Rules


• 2.7 – The Derivative As a Function
h  x  f  x  g  x ,
y  f  g , y  f  g   g  f 
f
g  f   f  g
y  , y 
g
g2
• 3.3 – Derivatives of Trig Functions

f   x   0 when f has horizontal tangents

f   x  is positive  f is increasing

y  sin x , y  cos x
y  cos x , y   sin x

f   x  is negative  f is decreasing

y  tan x , y  sec2 x
dy
 f   x   y 
dx
d2y
 f   x   y  2
dx
 If f is differentiable at
xa

 Important properties
1
1
1
 csc x,
 sec x,
 cot x
sin x
cos x
tan x
sin 2 A  cos 2 A  1
sin x
 tan x
cos x
, f is continuous at
 Continuous functions are not differentiable at:
a. hole or gap
b. vertical asymptote
c. corner or cusp
• 2.8 - What Does f  say about f ?

f   x  is positive  f is increasing

f   x  is negative  f is decreasing

f   x  is positive  f is concave up

• 3.4 – The Chain Rule

d n
 du 
u   n  u n 1   

dx
 dx 
2
 mess 

y   mess  ,

ye

y  a mess , y  a mess   mess  ln a


y  n  mess 
n
mess
, y  e
mess
n 1
  mess 
y   f g  x   f  g  x  
y  f   g  x    g   x 
dy dy du


dx du dx
• 3.7 – Derivatives of Logarithms





1
y  ln x , y 
x
1
x ln b
mess
y  ln  mess  , y 
mess
mess
y  logb  mess  , y 
mess  ln b
 f  x  g  x 
y  ln 

h  x


y  ln  f  x    ln  g  x    ln  h  x  
y  log b x , y 
y 
• 3.8 – Rates of Change In the Natural & Social
Sciences
 Average rate of change =
 Instantaneous rate of change =
 Linear approximation
L  x   f  a   f   a  x  a 
aka: Tangent line approximation
 Differentials
dy  f   x   dx
 Relative error: using differentials
If V  volume
dV
 instantaneous rate of change of volume
dr
dV used to calculate maximum error

f  g  h
 
f
g h
f  x  g  x
h  x
ln y  ln
f  x  g  x
h  x
ln y  ln  f  x    ln  g  x    ln  h  x  
differentiate
y  f  g  h
  
y
f
g h
 f  g  h 
y  y    
g h
 f
y 
dy
dx
• 3.9 – Linear Approximations and Differentials
 Logarithmic Differentiation
y
y
x
f  g   f  g  h  
    
h  f
g h 
..
3
Sample Problems :
1.
F  x 
8x
, find F   3
3  x2
2. Write the equation of the tangent to the
curve F  x  
8x
at x  3.
3  x2
3. Find f   a  given f  t  
5t  11
t4
4. The quantity (in pounds) of dog food
sold by Alpo, at a price of p dollars per
pound, is Q  f  p  .
9.
f  x  x  4
a. What is the domain of the function.
b. f   x  
c. What is the domain of the derivative?
a. What is the meaning of f   5  ?
10. Given the graph of y  f  x  , draw the
b. What are the units of f   5  ?
c. In general, will f   5  be positive or
graph of f   x  .
negative?
5. If a rock is thrown vertically upward on
the planet Magrathea, with a velocity of
20m/s, its height in meters after
seconds is given by H  20t  2.5t 2 .
a. When will the rock hit the surface?
b. With what velocity will the rock hit
11. Given the graph of f   x  ,
the surface?
6. Given f  x  h   f  x  
x2   x  h 
x2  x  h 
2
2
,
Find the slope of the tangent to
a. Over what intervals is f  x 
?
b. At what x values does f  x  have a
minimum?
y  f  x  at x  1.
7. Matching: The graphs of 3
derivatives are shown. Match the graph
of each function with the graph of its
derivative.
8. The graph of f  x  is shown. State the
x values at which f is not differentiable.
12. Let f  t  represent the temperature of
your patient at time t. f  2   99.5 F ,
f   2   1.5 , f   2   0.5 . Explain.
c Marcia Drost, October 19, 2015
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13. Sketch a function whose 1st and 2nd
b.
g  x    x  9  2 x  7 
c.
h  x 
d.
b
f  x   ax 2   c
x
e.
g  x   x  x  4
f.
h  x 
derivatives are always negative.
5
x3
4 x2  2 x 1
x
17. The equation of motion is s  t 3  9t ,
14. f  1  f   1  0,
a. v  t  
f   x   0, if x  1,
b. a  t  
f   x   0 if 1  x  2,
c. Find the acceleration when the
f   x   1, if x  2 ,
velocity is zero.
f   x   0 if 2  x  0,
18. Find the 1st and 2nd derivatives of
inflection point:  0,1
f  x   e2 x  x3
19. Find the points on the curve where the
tangent line is horizontal when
y  x3  24 x 2 .
20. Differentiate:
a. f  x   3e 2 x  x
15. Given f   x   x  e  x
 1 8
 3   x  5x2 
2
x 
x
b. g  x   
2
a. On what interval is f  x 
?
b. On what interval is f  x 
?
c. h  x  
x3
2  x2


d. f  x   x 2  3x e  r
21. f  3  1, g  3  2, f   3  4, g   3  3
Find h  3 .
a. h  x   4 f  x   5 g  x 
b. h  x  
16. Find the derivative:
a.
3
2
f  x   x 4  x 3  4 x  e3
2
3
f  x
1 g  x
c. h  x    f
g  x 
5
22. Differentiate:
a. y 
sin x
cos x  1
b. y  2 xe x  sin x
x
f
g
f’
g‘
2
3
4
5
-1
3
5
2
-2
6
4
-1
3
3
3
c. y  e  cos x
2x
2
23. Write the equation of the tangent to the
curve y  4 x  2cos x at x  0.
24. Write the equation of the tangent line to
8
at x  0.
y
sin x  2 cos x
cos x
25. f  60   2, f   60   3, g  x  
f  x
 
Find g  60
26. An elastic band is hung on a hook and a
mass is hung on the lower end of the
band. When the mass is pulled down
and released, it vibrates vertically. The
equation of motion s , in cm, and t  0
in sec, is given by s  4cos t  5sin t .
Find the velocity and acceleration at
time
.
27. Find the derivative of:

a. y  2  x 3

3
4

b. y  sin 2 a3  x 2

29. h  x    f


c. y  log 2 x 2  12 x

d. y  ln x3  e2 x


  x e  3 x 5  x  5 

e. y  ln 
2
3

 x  9  

 
f. y  ln 1  e 2 x
g. y 

3
ln  2x 
x2
31. A particle moves according to the law of
motion
s  f  t   3t 3  24t 2  72t
t  0, where is measured in seconds
and s in feet.
b. When is the particle at rest?
c. What is the total distance traveled in
32. Given the graph of the velocity function,
3
28. Find the equation of the tangent to the
curve y 

b. y  ln 50sin 2 x
the first 4 sec?
8 xcos x
 x2  2 
e. y  

 x5 
a. y  sin  5ln x 
a. Find the velocity at 3 sec.
c. y  2 x  e  kx
d. y  e
30. Differentitate:
4
at x  0. ,
1  e x
g  x  Find h  4  .
in sec, when is the article speeding
up?
6
33. If a ball is thrown vertically upward
with a velocity of 64ft/sec, then its
height after
sec is s  64t  16t 2 .
a. What is the maximum height
reached?
b. What is the velocity of the ball when
it first reaches 50 feet?
34. Find the linearization L  x  of the
function f  x   x3  5 x 2 at a  1 .
35. Use linear approximation to estimate
the value of 15.9 .
36. Find the differential of y 
u4
u 3