Math 131 Week in Review Sections 2.6-2.8
J. Lewis
1. Simplify each difference quotient for the given function at the given value of a.
f (a  h )  f (a )
i)
ii )
f ( x )  f (a )
xa
h
a)
f ( x )  mx  b
b)
f (x)  x
c)
f (x) 
d)
f (x) 
a 2
3
a 4
x
1
x
any a
a 1
2
2. Use your answer to problem 1 to find the derivative at a, f '(a), for each of
problem 1 a) - d).
3. Simplify the difference quotient
f (x  h)  f (x)
for each function of problem 1.
h
Use it to find f '(x) in each case. This is the limit definition of the derivative.
4. g ( x )  5 x 
3
3
x
2
 10
a) Use your answers to problem 3 and the linear property of derivatives to find g '(x) for.
b) Find the equation of the tangent line to the graph of g(x) at the point where x = 1.
5. f ( x )  3 x  18 x  12
2
a) Find
d
dx
f
( x )   f ' ( x ) using known derivatives and the linear property.
b) What is the slope of the tangent line to the graph of f at the point where x = 1?
Find this tangent line.
c) Find the equation of the tangent line to the graph of f at the point where x= -3.
d) Sketch the graph of f and each tangent line.
6. The position of an object traveling in a straight line is given by
s ( t )   16 t  64 t meters where t is in seconds.
2
a) Find v(t), the velocity at t seconds. Use known derivatives and the linear property.
What are the units?
b) Graph s(t) and v(t). Where is s(t) increasing and where is v(t) > 0? When does the
object stop and go back towards the initial position?
c) Find a(t), the acceleration. What are the units? Where is the velocity increasing?
decreasing? Where is the graph of s(t) growing steeper? less steep? Where is the object
accelerating ( where is the graph getting steeper) and which way is the object going at
this time? Where is the graph of s(t) concave down?
7. Repeat problem 5 for a) s ( t )  t  6 t
3
b)
2
s (t )  t  5t  t
2
3