In Class Exam 2 Review Math 131
1. What is the limit definition of the derivative with respect to x as a function?
Write the limit definition for the derivative of f ( x )  e x  x 2 .
2. Find the equation of the tangent line to f(x) at (a, f(a)).
a)
f (x)  4x
b)
f (x)  e
x
3 4
 2x  9
 ln x
a  16
a  1
3. Determine any and all x-values where the function is not differentiable.
2 5
a)
f (x)  x
b)
f (x)  (x
c)
|x |


2
f (x)  
x
 6 ( x  3 )  12

2
 9)
2 5
x 1
1 x  3
3  x
In part c, what change could be made to make f continuous at x=3? Then would it be
differentiable?
2
f (u )  u  2
4. h ( x )  f ( g ( x ))
g(x) at x=1.
Find the tangent line to h(x) at (1, h(1)).
y   x  4 is the tangent line to
5. Find f '(x) :
a)
x
f (x) 
e
b)
x
f (x) 
2x
2
1
 4x  7
x
c)
(x
ln 

2
3
 4) (
e
x
4
1
d)
f ( x )  cos  x
e)
f ( x )  sin
2
x


x
2
f )
f ( x )  tan( x )
g)
f ( x )  sec
5
(3 x )
6. If f(t) is distance and t is time and f(t) is increasing at an increasing rate then v(t) is
_______ and a(t) is _______________.
When v(t) has a maximum, a(t) is ______________ and the graph of f(t) has
____________.
If g '(x) is negative and is becoming more and more negative
then g is __________and _________________.
7. Shown is the derivative of f(x) . Determine the intervals where f(x) is
a) increasing, decreasing
b) concave up, concave down
c) Where does f (x) have a local max? a local min?
d) Where does f( x) have inflection points?
f '(x)
8. Use differentials or a linear approximation to approximate
a)
b)
3
3
66
63
9. An object travels in a straight line, east and west relative to its starting point.. Its
velocity at t seconds is
3
2
v ( t )  t  8 t  9 t  18
a) When is its eastward velocity the greatest?
b) When is its westward velocity the greatest?
c) For 0<t<6, when is the objects position the farthest east of start?
10. Use the limit definition of the derivative to find
d

3x
2
dx
i) What is the slope of the secant line from (x, f(x)) to (a, f(a))?
ii) Simplify until you can cancel x – a from the denominator.
iii) Find the limit as x approaches a.

 4 .