MATH 1910 Name________________
HW #5
Due: Monday, June 29
Write answers on a separate sheet of paper and staple them to this sheet. Show all
your work to receive full credit.
1. a.) Use the definition of the derivative to find
)(xf
′
for f(x) =
2 5
x
−
b.)
Use the answer from (a) to find the equation of the tangent line to the curve
f(x) =
2 5
x
−
at (3, 1). Include a sketch of the graph of f(x) with the tangent line.
2. Sketch a graph of a function f(x) that satisfies the following conditions. Label the
coordinate axes clearly.
a.)
( 2) (1) (3) 0
f f f
′ ′ ′
− = = =
b.)
f
(-2) = 3,
f
(0) = 1
c.)
(4) 1
f
′
= −
3. a.) Give an example of a function that is continuous at x = 4, but not differentiable at
x = 4 (don’t just sketch a graph, but provide an equation that satisfies the given
condition)
b.) Give an example of a function that is not continuous at x = 4, and therefore not
differentiable at x = 4 (don’t just sketch a graph, but provide an equation that satisfies the
given condition)
4. Let f(x) =
2
3 if 2
1 if 2
x x x
x x
+ − ≤
+ >
a.) Find a formula for
( )
f x
′
b.) Is f(x) differentiable at x = 2? If so, give the value of
(2)
f
′
. If not, explain
why not.
c.) Sketch the graphs of f(x) and
( )
f x
′
and clearly label the coordinate axes on
each graph.
5. Find the derivative of each function. Show all work, simplify where possible, and
write answers using positive exponents:
a.)
y =
5
6
11
x
x
−
b.) y =
33
3 (7 )
x x
−
c.)
y =
2
2
2 2
x
x x
+
− +
d.) y =
5
4 sin 2csc
x x x
−
e.) y =
2 3
tan (5 )
x
− f.) y =
4 2
12
(3 8 )
x x
−
