Homework for Math 16a - Analytic Geometry and Calculus - Fall 2008

This schedule is an approximation, and subject to change, depending on how things go during the semester. The schedule below indicates the sections of the text covered by homework due in each Tuesday section. We will post the actual assignments as the semester proceeds; we will only assign a subset of the questions from each section.

Sep 2: Questions on prerequisite material due

Simplify ((x+1)/(x^2 - 2*x - 3))/((x-3)/(x+1))

Simplify sqrt(a^4 - 4*a^2 + 4)
Note: "sqrt" means "square root"

Solve 3 = (6^y)/(2^y) for y

Simplify 2^(3*x)*4^(4-2*x)/ 16^(3-x)

Expand and simplify (2t+3)^3-(2t-3)^3

For f(x) = 3*x^2 - 2*x + 7, compute (f(a+h) - f(a))/h

answers, in pdf

Sep 9: Questions from Sections 0.1 - 0.6 due

Section 0.1: 18, 54

Section 0.2: 14, 32

Section 0.3: 14, 30, 34, 36

Let f(x)= 1+(1/(1+(1/(1+(1/x))))). Express f(a^2+2) as a rational function (quotient of two polynomials).

Section 0.4: 10, 24, 30, 36

Section 0.5: 18, 36, 46, 74, 94

Section 0.6: 20, 22

answers (all but Section 0.2, #32), in pdf

answers (Section 0.2, #32), in pdf

Sep 16: Questions from Sections 1.1 - 1.3 due

Section 1.1: 12, 24, 36, 52, 68

Section 1.2: 4, 22, 28, 30, 36

Use the Power Rule to find the derivative of (x^(-3/5))^2.

Let L be the tangent line to the curve f(x)=x^4-6 at (2,10). Let P be a line perpendicular to L and also passing through (2,10). Find the equation of P.

Section 1.3: 6, 30, 44, 54, 62, 68, 76, 88

answers, in pdf

Sep 23: Questions from Sections 1.4 - 1.6 due

Section 1.4: 18, 22, 36, 48, 52

Compute the limit of f(x) = ((x - 5*sqrt(x) + 6)/(x - 3*sqrt(x) + 2))^(1/5) as x -> 4

Section 1.5: 2, 8, 16, 24, 26, 34

Let c be a constant and let f(x) be defined to be f(x)=x^3 when x <= 1 and f(x)=c*x-c+1 when x > 1. For which numerical values of the constant c is the function f(x) continuous at x=1? For which numerical values of the constant c is the function f(x) differentiable at x=1?

Section 1.6: 24, 34, 42, 52, 56, 60

answers, in pdf

Sep 30: Questions from Sections 1.7 - 1.8 due

Section 1.7: 16, 26, 36, 38, 39, 40

Section 1.8: 8, 12, 18, 22

A toy rocket is launched from the ground. It accelerates straight upwards at 100 feet/sec/sec for 10 seconds, and then runs out of fuel. What is the maximum height it eventually reaches? How long after launch does it hit the ground? How fast is it going when it hits the ground? Hint: we did a similar problem in lecture.

Find the equation of the tangent line from (0,-16) to y=x^3.

Let C1 be the graph of the curve y=x^2, and let C2 be the graph of the curve y=-2x^2+4*x-3. Find the equation of a straight line y = m*x+b that is tangent to both C1 and C2. There are two such lines; find both of them. Hint: each line will intersect C1 and C2 at two different points (x1,y1) and (x2,y2). Try to find these points.

Let C1 be the graph of the curve y=x^2, and let C2 be the graph of the curve y=-2x^2+4*x-c, where c is a constant. For which value(s) of the constant c are there are exactly two different lines tangent to C1 and C2? For which value(s) of the constant c is there exactly one line tangent to C1 and C2? For which value(s) of the constant c are there no lines tangent to C1 and C2? Describe these three cases in terms of a geometric property of C1 and C2 (it may help to graph C1 and C2 in these 3 cases).

answers, in pdf

plots for the last question, in pdf

Oct 7: Questions from Sections 2.1 - 2.2 due

Section 2.1: 4, 6, 10, 12, 18, 28, 30, 40

Suppose that f(x) is differentiable and

increases when x < 8

decreases when 8 < x < 10

has an inflection point at x=9

increases when x > 10

Does f(x) have any relative maxima or minima? Where?

Let g(x) = f(x) + 83, where f(x) was defined in the last question. What properties does g(x) have, i.e. where does it increase, decrease, have any inflection points, relative minima, and relative maxima? Justify your answers.

Let h(x) = -g(x), where g(x) was defined in the last question. What properties does h(x) have, i.e. where does it increase, decrease, have any inflection points, relative minima, and relative maxima? Justify your answers.

Let f(x) = (x-1)^n, where n is a positive integer.

For which positive integer values of n does f(x) have a critical point? Where is it?

For which positive integer values of n does f(x) have a relative extremum? Where is it? Is is a relative maximum or minimum?

For which positive integer values of n does f(x) have an inflection point? Where is it?

Section 2.2: 2, 4, 6, 10, 30, 36, 42

answers, in pdf

plots, in pdf

Oct 14: Questions from Sections 2.3 - 2.5 due

Section 2.3: 6, 16, 20, 26, 32, 34

Section 2.4: 8, 18, 22, 28, 30, 34

Section 2.5: 4, 6, 8, 10, 12, 16, 18, 26, 30

answers, in pdf