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21A Homework 4
Due Thursday November 16
at your discussion.
Be sure to write your name, student ID and email clearly on your submission.
It is OK to work in teams on this homework but you must list
who you worked with so the TAs do not think you are copying
somebody else’s work. It is OK to use any reference (Google,
Wikipedia, Wolfram Alpha, books, research papers, your aunty
etc…) to solve these problems, but you MUST properly cite your
sources!
Question 1: Let f (x) = xx . Plot f on the interval (0, 1]. Suggest values for
limx→0+ f (x) and limx→0+ f 0 (x). Is f (0) defined? (Hint: You might want to
play around with Wolfram Alpha to answer this question. For example try
entering plot(xx , x = 0..1)). Now compute d(xx ).
Question 2: The limit of a product is the product of the limits. Is the
derivative of a product equal to the product of the derivatives? Is the relative
change1 in a product equal to the product of the relative changes? Explain
your answers with an example.
Question 3: Elasticity of demand. Given some quantity Q, its relative
change is dQ/Q. In economics Q might stand for how much of a certain
product is sold. If the price P of the product changes then we would expect
sales of the product to change. Therefore economists are often interested in
the relative change in Q per the relative change in price P . This is called the
elasticity of demand:
dQ/Q
.
=
dP/P
Would you typically expect the elasticity to be greater or less than one?
Compute in terms of P for the following “demand” curves
Q=
1
1
,
P2
Q = 200 − P/2 ,
Q = log P .
Remember that the (infinitesimal) relative change in f is df /f .
1
Question 4: Choose ten problems from the first 80 problems in Section 3.6
of Thomas’ Calculus and solve them. (Challenge yourself by choosing harder
problems, you are expected to be able to differentiate essentially any function
built from trigonometric functions, logarithms, exponentials and polynomial
functions of one variable.)
Question 5: The compound interest formula: an important limit that appears in many contexts and was first found by Bernoulli in a study of compound interest is
1 N
= e.
lim 1 +
N →∞
N
Lets try to see why this is true! In class we saw that (in base e)
log0 1 = 1 .
(i) Use the definition of the derivative and the above display to show that
log(1 + h)
= 1.
h→0
h
lim
(ii) Call N = 1/h. Fill in the blank:
When h −→ 0, N −→
.
(iii) Use parts i) and ii) to show
lim log
N →∞
h
1+
1 N i
= 1.
N
Hint: α log x = log xα .
(iv) Is the logarithm a differentiable function? If so, does this mean it is
continuous? What do you know about limits of continuous functions?
(v) Use exp(log(x)) = x and parts iii) and iv) to show that Bernoulli’s
compound interest result is true.
Question 6: Explain why
d cos−1 u = − √
2
du
.
1 − u2
Include a picture in your answer.
Question 7 A ladder leans again the side of a building. The base of the
ladder sits in an oil slick which causes the ladder to slide down the wall. Find
a formula relating the the rate of change of the angle formed by the ladder
and the ground in terms of the rate of change of the distance of the base of
the ladder from the wall. (Hint: First draw a picture. Introduce symbols
labelling any quantities that you think are relevant for the problem.)
Question 8: A rectangle has fixed perimeter of length 100’. The length of
the rectangle is increasing at 1’/second. At what rate does the area of the
rectangle change? For which lengths is this increasing or decreasing?
Question 9: A 20’ tall giraffe stands 10’ away from the base of a tall lamppost at night and casts a 5’ shadow. The giraffe then walks away from to
lamppost at 1’/second. At what rate does the giraffe’s shadow lengthen?
Question 9: Partial derivatives: Let F : R3 → R. This is a function of
three real variables F (x1 , x2 , x3 ). Demonstrate using some examples of F
why the infinitesimal change dF in F when all three variables x1 , x2 , x3 are
(respectively) changed by dx1 , dx2 , dx3 must take the form:
dF (x1 , x2 , x3 ) = f1 (x1 , x2 , x3 )dx1 + f2 (x1 , x2 , x3 )dx2 + f3 (x1 , x2 , x3 )dx3 .
(This question is asking you to dream up functions like F = x1 x2 x3 , compute dF and then look for a pattern.) The mathematical notation for the
three functions f1 , f2 , f3 is ∂F/∂x1 , ∂F/∂x2 , ∂F/∂x3 . Look up the “partial
derivative” (in your textbook or online, but do cite your source). Summarize
what you find, and discuss why it makes sense for the above computation.
Question 10: What to do when the
√ linearization1 vanishes: Find the linearization of the function f (x) = 1 + x − 1 − 2 x at x = 0. Sketch the
graph of both f (x) and its linearization. Now find the linearization L(x) of
the function f 0 (x) at x = 0. Now make a careful plot of the functions f (x)
and 21 xL(x) on an interval around x = 0.
3
…
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