MATH 136 CALCULUS 1
FALL 2011
TEST TWO REVIEW
8.
Suppose f (x ) has the following graph. On this same graph,
sketch f ' ( x ) . If you're careful and thoughtful, you should be able to
get a very accurate graph of f ' ( x ) .
Here’re some problems you should be able to answer for the test on
Friday. On all problems, you should be prepared to provide explanation
if asked. (Selected solutions are included in parentheses – but I can’t
guarantee they’re all right.)
1. State the limit definition of the derivative and draw a picture which
shows where this definition comes from.
2. Using the limit definition of the derivative, find f ' ( x) for the
following functions.
a) f ( x)  x  x
2
b) f ( x) 
4x  1
1
c) f ( x) 
x3
3. Using more efficient methods, find f ' ( x) for the following functions.
a) f ( x)  x12  8x 3
b) f ( x)  ( x 2  sin( x)) x  1
x2
d) f ( x)  ln(arctan( x))
x4  3
5
3
4e x 2 x 3 x
e) f ( x)  2
3x cos( 2x )  x ln( 2 x)
c) f ( x) 
9. At t=0, a hot water faucet is turned on and the temperature of the
water coming out of the faucet is described by H(t), where t is
measured in minutes. (You might want to make a sketch of what you
think H(t) might look like in order to help you answer these
questions.)
Explain, in everyday terms, what it would mean if H’(t)>0.
Which would have a larger value, H’(.25) or H’(8)?
Explain, in everyday terms, when and why H’(t) will be negative.
X. Find the tangent line to f ( x)  4 x 2  2 which passes through the
point (1, 1).
4. Find the equation of the tangent line to f ( x)  x cos( x)  e x at
x  0. ( y  x 1)
2
5. Suppose the position of an object is given by
x(t )  (2  t ) cos(t )  sin( t ) . Find the first two time values at which
the object is stationary. ( t  2 ,2 )
6. Find all of the x-values at which the graph of
g ( x)  2 x 3  3x 2  36 x  12 has a horizontal tangent. (x=-2, 3)
MATH 136 CALCULUS 1
FALL 2011
TEST TWO REVIEW
8.
Suppose f (x ) has the following graph. On this same graph,
sketch f ' ( x ) . If you're careful and thoughtful, you should be able to
get a very accurate graph of f ' ( x ) .
Here’re some problems you should be able to answer for the test on
Friday. On all problems, you should be prepared to provide explanation
if asked. (Selected solutions are included in parentheses – but I can’t
guarantee they’re all right.)
1. State the limit definition of the derivative and draw a picture which
shows where this definition comes from.
2. Using the limit definition of the derivative, find f ' ( x) for the
following functions.
a) f ( x)  x  x
2
b) f ( x) 
4x  1
1
c) f ( x) 
x3
3. Using more efficient methods, find f ' ( x) for the following functions.
a) f ( x)  x12  8x 3
b) f ( x)  ( x 2  sin( x)) x  1
x2
d) f ( x)  ln(arctan( x))
x4  3
5
3
4e x 2 x 3 x
e) f ( x)  2
3x cos( 2x )  x ln( 2 x)
c) f ( x) 
9. At t=0, a hot water faucet is turned on and the temperature of the
water coming out of the faucet is described by H(t), where t is
measured in minutes. (You might want to make a sketch of what you
think H(t) might look like in order to help you answer these
questions.)
Explain, in everyday terms, what it would mean if H’(t)>0.
Which would have a larger value, H’(.25) or H’(8)?
Explain, in everyday terms, when and why H’(t) will be negative.
X. Find the tangent line to f ( x)  4 x 2  2 which passes through the
point (1, 1).
4. Find the equation of the tangent line to f ( x)  x cos( x)  e x at
x  0. ( y  x 1)
2
5. Suppose the position of an object is given by
x(t )  (2  t ) cos(t )  sin( t ) . Find the first two time values at which
the object is stationary. ( t  2 ,2 )
6. Find all of the x-values at which the graph of
g ( x)  2 x 3  3x 2  36 x  12 has a horizontal tangent. (x=-2, 3)