This document provides 3 problems to practice differentiation rules and implicit differentiation to justify the power rule for non-positive integer powers. Students are asked to use derivative rules and implicit differentiation to show that the power rule holds for rational powers, irrational powers, and negative powers. The problems also require showing work and using proper notation for full credit.
Written Homework 2, Math122B section 5.SHOW ALL WORK REAS.docx
1. Written Homework 2, Math122B section 5.
SHOW ALL WORK/ REASONINGS AND USE PROPER
NOTATION FOR
FULL CREDIT.
During lecture, we proved the power rule for xn where n is a
positive integer. We generalized
without proof that the power rule will persist for any real
number n. In this homework, we will
use derivative rules and implicit differentiation to justify the
power rule for non-positive-integer
power.
1 If the power of the power function is a rational number, we
can write the power as
m
n
. Let
y = x
m
n . Then yn = xm. Use implicit differentiation to compute
dy
dx
2. and confirm that the
power rule holds for rational power.
2 Suppose the power n is an irrational number. Let y = xn where
n is irrational. Apply the
natural log to the equation to ”pull down” the exponent n. Then
use implicit differentiation
to compute
dy
dx
and again, confirm that the power rule holds for irrational
power.
3 The remaining case to check is negative power. Let y = x−n
where n > 0. Use chain rule
to compute
dy
dx
and that the power rule holds for negative power as well.
1
Math 122B
Name ________________________
3. Section _______
3.1-3.6: DIFFERENTIATION PRACTICE
Find the indicated derivative in each case. You should try to
simplify your answers if you can. Try quotient rule on problems
1, 10, 17, and 18.
1.
()
ft
¢
for
3
()
1
t
ft
t
=
+
2.
()
fx
¢
for
2
3
1
()
14. x
=
.
2. Determine if
(())
yhgx
=
is increasing or decreasing at
3
x
=
.
3. Find the equation of the tangent line to
(())
yfgx
=
at
2
x
=
.
4. Find
17. Application Problems:
1.
The quantity, q, of ice cream cones sold depends on the selling
price, p, in dollars, so we can write q = f (p). You are given that
f (5) = 1,000 and f ′(5) = –50.
(a)
What do f (5) = 1,000 and f ′(5) = –50 tell you about the sales
of ice cream cones?
(b)
The total revenue, R, earned by the sale of ice cream cones is
given by R = pq. Find .
(c)
What is the sign of ? If the ice cream cones are currently
selling for $5, what happens to revenue if the price is increased
to $6?
2.
A smokestack deposits soot on the ground with a concentration
inversely proportional to the square of the distance from the
stack. With two smokestacks 20 miles apart, the concentration
of the combined deposits on the line joining them, at a distance
x from one stack, is given by
where k1 and k2 are positive constants which depend on the
quantity of smoke each stack is emitting. For what value of is
?
_1046872686.unknown
