MATH 137 Exam Review: Sequences, Limits, Continuity, Derivatives and Taylor Polynomials

Terms in this set (29)

1. Formal definition: a sequence {a_n} converges to LFor every epsilon > 0 there exists N such that for all n > N, |a_n - L| < epsilon.
2. State the Monotone Convergence Theorem.Every bounded monotonic sequence converges. An increasing sequence bounded above converges to its least upper bound.
3. Does a_n = (-1)^n converge?No. The terms alternate between -1 and 1, so there is no single limit; the sequence is bounded but divergent.
4. Squeeze TheoremIf g(x) <= f(x) <= h(x) near a and lim g(x) = lim h(x) = L as x -> a, then lim f(x) = L as x -> a.
5. Evaluate lim (x -> 0) of x^2 sin(1/x).0. Since -x^2 <= x^2 sin(1/x) <= x^2 and both bounds go to 0, the Squeeze Theorem gives 0.
6. Epsilon-delta definition of lim (x -> a) f(x) = LFor every epsilon > 0 there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.
7. What is the fundamental trig limit?lim (x -> 0) sin(x)/x = 1 (x in radians).
8. Evaluate lim (x -> 3) (x^2 - 9)/(x - 3).Factor: (x - 3)(x + 3)/(x - 3) = x + 3 for x not equal to 3, so the limit is 6.
9. Three conditions for f to be continuous at x = af(a) is defined, lim (x -> a) f(x) exists, and the limit equals f(a).
10. Name the three types of discontinuity.Removable (limit exists but differs from f(a) or f(a) undefined), jump (left and right limits exist but differ), and infinite/essential (a one-sided limit is infinite or does not exist).
11. State the Intermediate Value Theorem.If f is continuous on [a, b] and k is any value between f(a) and f(b), then there is some c in [a, b] with f(c) = k.
12. How do you use the IVT to show x^3 + x - 1 = 0 has a root in [0, 1]?f(0) = -1 < 0 and f(1) = 1 > 0, and f is a continuous polynomial, so by the IVT there is c in (0, 1) with f(c) = 0.
13. Limit definition of the derivativef'(a) = lim (h -> 0) [f(a + h) - f(a)] / h, provided the limit exists.
14. Does differentiability imply continuity? Does the converse hold?Differentiable at a implies continuous at a. The converse is false: f(x) = |x| is continuous at 0 but not differentiable there.
15. Product rule and quotient rule(fg)' = f'g + fg'. (f/g)' = (f'g - fg') / g^2, where g is not 0.
16. Chain rule; differentiate sin(x^3).d/dx f(g(x)) = f'(g(x))·g'(x). So d/dx sin(x^3) = cos(x^3)·3x^2.
17. Derivatives of e^x, ln x, a^xd/dx e^x = e^x; d/dx ln x = 1/x for x > 0; d/dx a^x = a^x·ln a.
18. Derivatives of tan x, arcsin x, arctan xd/dx tan x = sec^2 x; d/dx arcsin x = 1/sqrt(1 - x^2); d/dx arctan x = 1/(1 + x^2).
19. Find dy/dx for x^2 + y^2 = 25 by implicit differentiation.2x + 2y·dy/dx = 0, so dy/dx = -x/y.
20. When should you use logarithmic differentiation? Differentiate y = x^x.When a variable is in both base and exponent, or for messy products. ln y = x ln x, so y'/y = ln x + 1, giving y' = x^x (ln x + 1).
21. Linear approximation of f near x = aL(x) = f(a) + f'(a)(x - a). Example: sqrt(4.1) is about 2 + (1/4)(0.1) = 2.025.
22. State the Mean Value Theorem.If f is continuous on [a, b] and differentiable on (a, b), there is c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a).
23. What is a consequence of the MVT about functions with zero derivative?If f'(x) = 0 for all x in an interval, then f is constant on that interval.
24. First derivative test for increasing/decreasingIf f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing. A sign change of f' at a critical point indicates a local max or min.
25. State L'Hospital's Rule.If lim f(x)/g(x) is of the form 0/0 or infinity/infinity, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.
26. nth degree Taylor polynomial of f centred at aT_n(x) = sum from k = 0 to n of f^(k)(a)/k! · (x - a)^k.
27. Write the degree 3 Taylor polynomial of e^x and sin x centred at 0.e^x: 1 + x + x^2/2 + x^3/6. sin x: x - x^3/6.
28. Taylor's Remainder Theorem (Lagrange form)f(x) - T_n(x) = f^(n+1)(c)/(n+1)! · (x - a)^(n+1) for some c between a and x; bounding f^(n+1) bounds the error.
29. Newton's method iteration formulax_(n+1) = x_n - f(x_n)/f'(x_n). It approximates a root of f by following tangent lines; it may fail if f'(x_n) is near 0.