hw4.pdf - Math 2A Single Variable Calculus Homework...

Math 2A Single Variable Calculus Homework Questions 4
4 Applications of Derivatives
4.1 Maximum and Minimum Values
1. Find the critical points of f
(
x
) =
x
3
+
6 x
2
-
15 x .
2. Find the critical points of h
(
p
) =
p
-
1
p 2 + 4
.
3. Find the critical points of g
(
θ
) =
4 θ
-
tan θ .
4. Find the absolute maximum and minimum values of f ( x ) = 5 + 54 x - 2 x
3
on the interval [ 0, 4 ] .
5. Find the absolute maximum and minimum values of f
(
x
) =
x
3
-
6 x
2
+
5 on the interval
[ -
3, 5
]
.
6. Find the absolute maximum and minimum values of f ( x ) =
x
x 2
-
x
+
1
on the interval [ 0, 3 ] .
7. Find the absolute maximum and minimum values of f
(
t
) =
3
√
t
(
8
-
t
)
on the interval
[
0, 8
]
.
8. An object with weight W is dragged along a horizontal plane by a force acting along a rope
attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the
force is
F =
μ W
μ sin θ + cos θ
,
where μ is a positive constant called the coefficient of friction and 0 ≤ θ ≤ π /2. Show that F is
minimized when tan θ
=
μ .
9. Show that 5 is a critical number of the function
g ( x ) = 2 + ( x - 5 )
3
,
but that g does not have a local extreme value at x = 5.
4.2 The Mean Value Theorem
1. Let f ( x ) = tan x . Show that f ( 0 ) = f ( π ) but that there is no number x in ( 0, π ) such that
f
0
( c ) = 0. Why does this not contradict Rolle’s Theorem?
2. Verify that the function f
(
x
) =
x
3 -
3 x
+
2 satisfies the hypotheses of the Mean Value Theorem
on the interval
[ -
2, 2
]
, and find all numbers c that satisfy the conclusion of the Mean Value
Theorem.
3. Verify that the function f ( x ) = 1/ x satisfies the hypotheses of the Mean Value Theorem on the
interval
[
1, 3
]
, and find all numbers c that satisfy the conclusion of the Mean Value Theorem.
4. Let f
(
x
) =
2
- |
2 x
-
1 | . Show that there is no value of c such that f
(
3
) -
f
(
0
) =
f
0 (
c
)(
3
-
0
)
.
Why does this not contradict the Mean Value Theorem?
1

5. Show that the equation 2 x - 1 - sin x = 0 has exactly one real root.
6. Show that the equation x
4
+ 4 x + c = 0 has at most two real roots.
7. Suppose that 3
≤
f
0
(
x
) ≤
5 for all values of x . Show that 18
≤
f
(
8
) -
f
(
2
) ≤
30.
8. Suppose that f and g are continuous on
[
a , b
]
and differentiable on
(
a , b
)
. Suppose also that
f
(
a
) =
g
(
a
)
and that f
0
(
x
) <
g
0
(
x
)
for a
<
x
<
b . Prove that f
(
b
) <
g
(
b
)
. ( Hint: Apply the Mean
Value Theorem to the function h = f - g. )
9. At 2 pm a car’s speedometer reads 30 mi/h. Ten minutes later it reads 50 mi/h. Prove that at
some time between 2 and 2.10 pm the acceleration of the car is exactly 120 mi/h
2
.
10. A number a is called a fixed point of a function f if f ( a ) = a . Prove that if f
0
( x ) 6 = 1 for all real
numbers x then f has at most one fixed point.
4.3 How Derivatives Affect the Shape of a Graph
1. Let