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? .?. The equation x2 = x + x + . + x is valid only when x
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 domain of f. We call f(c) the maximum value of f on D. We sa
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 all x in [a,b]. That is, every continuous function whos
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 differentiable on the open interval (a,b). (We assume a < b.)
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 neighborhood of c (but not necessarily at c itself). (a)
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: We say y = L(x) := mx + b is a slant asymptote to y = f
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) and show that it gets arbitrarily close to the line y
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) Critical points: P = 2 2/x2 which is undefined up at
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 time to argue that the optimum solution should have x =
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 (,), and that this value is the unique real number satisfying
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 exp(x) + 6. #2: "If f is one-to-one and differentiable,
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 near a except possibly at a itself. Then limxa f(x)/g(x
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/b = ln a ln b. It says nothing if ln a is undefined an
[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 explicitly in the form f(x); it can be enough to write an al
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 138139: 1. "If f is continuous at a, then f is differe
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 function in the vicinity of x = a, then: for x a, the (u
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 hemispherical dome with diameter 50 m. Solution: For 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 = sqrt(2) or r = sqrt(2)sqrt(2) works, but the proof w
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 domain of f with x1 < x2, then since f is increasing, f
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 interval I, then ln f(x) is differentiable with derivat
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 interest, for a total of $100 times 1.12 = $112. If your mone
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(1x2) for x in the interval . .?. (1,1). If f(x) = cos1
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 most general antiderivative of f(x) = 1/x on the set S =
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 (concluded) Last time we asked "Does the function f(x) = |x| hav
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 x) (ex) is .?. (1/2) (sin x) (ex) (1/2) (cos x) (ex).
[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
direction on the horizontal axis goes to the right, the graph
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) Suppose f(x) is increasing on
cfw_x: x < 0 and increasin
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 < a +
x a > and x a <
|x a| <
x lies (strictly) within
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 simple theorems about
limits.
Suppose that limxa f(x) e
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 more modest claim?: If f(a) is defined and
limxa f(x) exi