Hand out practice exam. Paradox: Write x2 = x + x + . + x (x added to itself x times). Differentiating both sides, we get 2x = 1 + 1 + . + 1 (1 added to itself x times), or 2x = x. What's going on? .?
Section 4.1: Key concepts? .?. Absolute (or global) vs. relative (or local) maxima and minima, critical numbers A function f with domain D has a global maximum at c if f(c) f(x) for all x in the domai
Section 4.1 (continued): Theorem 3 (Extreme Value Theorem): If f is continuous on the closed interval [a,b], then there exist c and d in [a,b] such that f(x) f(c) for all x in [a,b] and f(x) f(d) for
Section 4.2: The Mean Value Theorem (continued) Mean Value Theorem: Let f be a function on the interval [a,b] (with a < b) that satisfies 1. f is continuous on the closed interval [a,b] 2. f is differ
Section 4.3: Derivatives and the shapes of graphs (continued) First Derivative Test: Suppose that c is a critical number of a continuous function f, and suppose that f (x) is defined for all x in a ne
Section 4.4: Curve Sketching To sketch a curve y = f(x), first identify: A. Domain B. Intercepts C. Symmetry (and periodicity) D. Asymptotes (vertical, horizontal, or slant) (See exercises 4.4.47-50:
More about slant asymptotes: Consider the hyperbola x2 y2 = 1:
2
1
0
-1
-2 -2 -1 0 1 2
To see that the hyperbola x2 y2 = 1 has the line y = x as a slant asymptote, take the upper branch y = sqrt(x2 1)
Section 4.5, continued: Problem: Which rectangle with area 1 has smallest perimeter? Goal: Minimize P = 2x + 2y subject to xy = 1 Write y in terms of x: .?. y = 1/x P(x,y) = P(x) = 2x + 2/x (for x > 0
Section 4.5, concluded: Example 5: Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. Maximize A = 2xy subject to x2 + y2 = r2 with x,y 0. We used symmetry last
Last time we started on the problem of approximating the minimum value of the function x2 sin x. We showed that there is a unique value x* for which this function achieves its global minimum on (,), a
Chapter Review true/false questions on page 195 #1: "If f is one-to-one, with domain R, then f 1(f (6) = 6." .?. Answer: True. But note that f (f 1(6) need not be defined; e.g., let f(x) = exp(x) or e
Section 3.7: L'Hospital's Rule (concluded) Recall the formal statement of L'Hospital's Rule: Suppose limxa f(x) = 0 and limxa g(x) = 0. Suppose furthermore that f and g are differentiable and g (x) 0
Paradox: 0 = ln 1 = ln (1)/(1) = ln (1) ln (1) = undefined! What's going on? . .?. The rule ln a/b = ln a ln b only says that if ln a and ln b are defined (that is, if a and b are positive), then ln a
[Hand back homework and hand out practice test solutions.] What's the main idea of section 2.6? . Implicit differentiation: To differentiate y with respect to x, you don't always need to write y expli
Questions about the midterm exam? Reminder: Your cheat sheet must be WRITTEN or TYPED by you. The exam will cover up through (and including) section 2.5 (the chain rule). True-false questions on pages
Hand back homework, collect section notes Section 2.8: Linear approximation and differentials Main idea? .?. Derivatives are good for finding approximate values of functions If f is a differentiable f
Section 2.8: Linear approximation and differentials (concluded): Section 2.8, Problem 24: Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispher
Puzzle from Monday's lecture: Does there exist an irrational number r such that rsqrt(2) is rational? Hint: The answer is in Wednesday's lecture (sort of!). Solution: We can easily prove that either r
Section 3.2: Inverse functions and logarithms (concluded) Fact: If f is an increasing function then f is one-to-one. (Note: Here "increasing" means "strictly increasing".) Proof: If x1, x2 are in the
Last time we saw one way to compute the derivative of xx with respect to x. Another method we can apply is logarithmic differentiation: (d/dx) ln f(x) = f (x) / f(x) That is, if f(x) > 0 on some open
Section 3.4: Exponential growth and decay (concluded). Interest If you put $100 in the bank and it's compounded yearly with 12% annual interest, after 1 year you get your original $100 plus $12 intere
Section 3.5: Inverse trig functions (concluded). Last time we defined arcsin and arccos (also written as sin1 and cos1), and showed that (d/dx) arcsin x = 1 / sqrt(1x2) and (d/dx) arccos x = 1 / sqrt(
Section 4.7: Antiderivatives (continued) Recap of last time: The function f(x) = 1/x doesn't have an antiderivative on R, because . .?. f(x) isn't even defined at x = 0. We showed last time that the m
For Wednesday: do true/false questions for chapter 4. Also, does there exist a twice-differentiable function f on R satisfying f (x) < 0 and f (x) > 0 for all x? Section 4.7: Antiderivatives (conclude
Note: The exams may not have exactly three questions per day. If there are more questions, they'll be easier individually! Sometimes finding antiderivatives is tricky: An antiderivative of f(x) = (sin
[Collect section-summaries]
Remember, your section summaries must have YOUR
NAME.
Page 17: Note that the horizontal shift rule is easy to get
backwards. (Since its a plus sign, and the positive
direct
Whos looked at the solutions to the first assignment?
.?.
.?.
Was it helpful?
Define The function f is increasing on the set S to mean
For all x1 < x2 in S, f(x1) < f(x2).
Problem D (worth 6 points) S
Absolute value and distance (preparatory for section 1.3):
For any positive number , the following are equivalent (if
x satisfies ANY of them, it must satisfy ALL of them):
x (a , a + )
x > a and x <
Section 1.4: Laws governing limits
Main idea of section 1.4:
.?.
Just as you can build complicated functions from simple
functions, you can compute limits of complicated
functions using a dozen or so
Section 1.3: The notion of limit (continued)
True or false?: If f(a) is defined, limxa f(x) = f(a).
.?.
.?.
You could have a function f for which f(a) is defined but
limxa f(x) isnt.
How about this mo