Derivative review PDF

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MTH 202

Prof. Sundstrom Derivative Review Problems

1. The graph to the right contains the graph of a function y  f  x  and several points and lines. For each of the quantities below, state whether it is represented on the graph by a length or by a slope. Then using the points represented by letters, make clear exactly which length or slope represents the quantity. Note: The letters, P, Q, R, etc. represent points of intersections of lines and/or curves. The line QU is tangent to the graph of y  f ( x) at Q and the line VT is tangent to the graph of y  f ( x) at T.

a.

f ( a  h)  f ( a)

c. lim

f  a  h  f  a

h 0

h

b.

f ( a  h)  f ( a) h

d.

f ( a)  h

2. Assume that we are given f  x  and g  x  . Write formulas for the derivatives of each of the following functions by using the product rule, the quotient rule, or the chain rule. p x   f x  g x 

q x  

f x 

g x

p  x  

q  x  

h  x   f  g  x 

h  x  

k  x   g  f  x 

k  x 

3. Write formulas for the derivatives of each of the following functions. The last three derivatives should be written in terms of u  x and u  x  .

MTH 202 – Derivative Review f  x   e x

f   x 

4

sin  2 x

g  x 

3x

page 2

g x 

2

h  x  x3 ln  3x2  2

h x 

3 u  x

k  x 

G x  ln 5u  x  3

G x 

k  x 

H  x 

sin  u  x 

H  x  

x

4. The graph to the right is a graph of the equation ln  xy   2 x .

The point 1, e 2  is shown on the graph. a. Use implicit differentiation to determine a dy . (Your answer should be formula for dx a formula in terms of x and y.)

b. Use your result from Part (a) to determine the equation of the line tangent to the graph of the equation ln  xy   2 x at the point 1, e 2  .

MTH 202 – Derivative Review

page 3

5. For time t measured in seconds, the graph to the right shows the vertical velocity v t  (in feet per second) of an object. (The upward direction is the positive direction and the downward direction is the negative direction.) The object started from ground level [ s  0  0 , where s  t  is the vertical altitude of the object in feet].

The graph consists of two straight lines and two quarter circles of radius 20. a. Over what intervals is the vertical altitude s  t  increasing? b. At what time was the greatest vertical altitude attained? c. What is the value of



100

0

v t  dt ?

d. What was the average vertical velocity of the object between 0 and 100 seconds?...