hw3.pdf - Math 2A Single Variable Calculus Homework...

Math 2A Single Variable Calculus Homework Questions 3
3.1 Derivatives of Polynomials and Exponential Functions
1. Differentiate f ( x ) = π
2
.
2. Differentiate h ( x ) = ( x - 2 )( 2 x + 3 ) .
3. Differentiate y
=
x
5/3 -
x
2/3
.
4. Differentiate v
=
√
x
+
1
3
√
x
2
.
5. Differentiate y
=
4
√
t
-
4 e
t
.
6. Differentiate k ( r ) = e
r
+ r
e
.
7. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the
pressure P of the gas is inversely proportional to the volume V of the gas.
(a) Suppose that the pressure of a sample that occupies 0.106 m
3
at 25
◦
C is 50 kPa. Write V as
a function of P .
(b) Calculate
d V
d P
when P
=
50 kPa. WHat is the meaning of the derivative? What are its units?
8. Find the n th derivative of each function by calculating the first few derivatives and observing
the pattern that occurs:
(a) f
(
x
) =
x
n
, (b) f
(
x
) =
1
x
.
3.2 The Product and Quotient Rules
1. Find the derivative of f ( x ) = ( 1 + 2 x
2
)( x - x
2
) in two ways: via the Product Rule and by
multiplying out first. Do your answers agree?
2. Differentiate F ( y ) =
1
y
2
-
3
y 4
( y + 5 y
3
) .
3. Differentiate J
(
v
) = (
v
3 -
2 v
)(
v
-
4 +
v
-
2 )
.
4. Differentiate y =
t
(
t
-
1
) 2
.
5. Differentiate y = A +
B
x
+
C
x 2
.
6. Differentiate f
(
x
) =
ax + b
cx
+
d
.
7. Find an equation for the tangent line to the curve y
=
2 x
x + 1
at (1,1).
8. If g ( x ) =
x
e x
, find g
( n )
( x ) .
9. The curve y
=
x /
(
1
+
x
2 )
is called a serpentine.
(a) Find an equation of the tangent line at the point ( 3, 0.3 ) .
1

(b) Illustrate part (a) by graphing the curve and the tangent line.
10. A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric
(in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write
q = f ( p ) . The total revenue earned is then R ( p ) = p f ( p ) .
(a) What does it mean to say that f
(
20
) =
10, 000 and f
0
(
20
) = -
350?
(b) Assuming the values in part (a), find R
0
( 20 ) and interpret your answer.
11. At what numbers is the following function differentiable?
g
(
x
) =





2 x if x ≤ 0,
2 x - x
2
if 0 < x < 2,
2
-
x if x
≥
2.
Give a formula for g
0
and sketch the graphs of g and g
0
.
12. Sketch the parabolas y
=
x
2
and y
=
x
2 -
2 x
+
2. Do you think there is a line that is tangent to
both curves? If so, find its equation. If not, why not?
3.3 Derivatives of Trigonometric Functions