ECE/CS 313 Problem Set # 1 Fall 2002
Assigned: 8/28/02 Due: 9/4/02
Calculus tune up
Assigned reading: Review a calculus book! Also read Ross Chapter 1.1–1.4, Chapter 2.1–2.5.
Noncredit exercises: Chapter 1, problems 1-5,7,9; Theoretical exercises 4,8,13; Self-test problems 1-15;
Chapter 2: problems 3,4,9,10, 11-14; Theoreitical exerecises 1-3,6,7,10,11,12,16,19,20; Self-test problems 1-8.
About the credit problems (below): Calculus, a prerequisite of this course, will be used mainly in
the second half of the semester. The parts of calculus primarily needed are integration and differentiation,
Taylor series, l’Hospital’s Rule, integration by parts, and double integrals (setting up limits of integration
and change of variables such as in rectangular to polar coordinates). This problem set will help you review
these topics, and identify areas in which you may need additional review.
1. Geometric series
(a) Prove that 1 + x+x2+...+xn−1=1−xn
1−xfor x6= 1 and integer n≥1.
(b) Assuming |x| ≤ 1, find the sum 1 + x+.... (By definition, this sum is the limit of the sum 1 + x+x2+
...+xn−1as n→ ∞.)
2. Inverse function
Let fbe the function defined by f(x) = 2
ex+1 for −∞ <x<∞. Let gdenote the inverse function of f, so
that g(f(x)) = xfor all x. Find g. (Hint: The domain of gcan be smaller than the whole real line. Draw a
picture.)
3. Extrema of functions
(a) Find the maxima (plural) of f(x) = x4(1.001)−x2on the real line. (b) Does the function f(x) =
x4(1.001)x2have a maximum value at some finite x? Explain.
4. Some definite integrals
Find the values of the following definite integrals:
(a) Z1
−1
|x|dx; (b) Z1
0
x(1 −x2)11dx; (c) Z1
0
x2exp(−x)dx;
(d) Z∞
−∞
x3exp(−x2/2) dx. (Hint: Plotting the integrand might give you some clues.)
5. Derivatives and integrals
Let d
dxf(x) = g(x),−∞ <x<∞. Which of the following statements is true for all x? (Here, Cdenotes
an arbitrary constant.)
(a) d
dx f(−x) = −g(−x) (b) d
dx f(x2/2) = xg(x2/2) (c) d
dx exp f(x2)=g(x2) exp f(x2)
(d) Rg(−x)dx =f(−x) + C(e) Rg(x2/2) dx =f(x2/2)
x+C(f) Rg(x)
f(x)dx = ln(f(x)) + C
6. Double integrals
Evaluate the following definite two-dimensional integrals over the specified domains of integration:
1. f(x, y) = max(x, y), over the region {(x, y) : 0 ≤x≤1,0≤y≤1}.
2. f(x, y) = (x2+y2)−4, over the region {(x, y) : x2+y2>1}.
7. An application of Taylor’s approximation
(a) Find the degree one Taylor polynomial P(x) = f(0) + f0(0)xfor the function f(x) = ln(1 + x).
(b) Find a number Mso that |f00(x)| ≤ Mfor |x| ≤ 0.5.
(Hint for next part: It follows by Taylor’s Inequality that |f(x)−P(x)| ≤ Mx2
2! for |x| ≤ 0.5.)
(c) Given any real value a, find a simple expression for limn→∞ nln(1 + a
n).
(d) Find a simple expression for limn→∞(1 + a
n)n.(Hint: (1 + a
n)n= exp(ln((1 + a
n)n)).)