M 408C Exam Review: Limits, Continuity, Derivatives, Mean Value Theorem and Integration

Terms in this set (30)

1. When does lim as x→a of f(x) exist?When the left-hand limit and the right-hand limit both exist and are equal. The value of f(a) itself does not matter for the limit.
2. Evaluate lim as x→2 of (x^2 − 4)/(x − 2).Factor: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2, so the limit is 4.
3. State the squeeze theorem and give its classic use.If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L. It shows lim as x→0 of x^2 sin(1/x) = 0, since −x^2 ≤ x^2 sin(1/x) ≤ x^2.
4. Two key trigonometric limitslim as x→0 of sin x / x = 1 (x in radians), and lim as x→0 of (1 − cos x)/x = 0.
5. Three conditions for f to be continuous at x = af(a) is defined, lim as x→a of f(x) exists, and the limit equals f(a).
6. State the Intermediate Value Theorem.If f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for some c in (a, b). It is often used to show a root exists when f(a) and f(b) have opposite signs.
7. Removable vs jump vs infinite discontinuityRemovable: the limit exists but differs from f(a) or f(a) is undefined (a hole). Jump: the one-sided limits exist but differ. Infinite: f grows without bound near a (vertical asymptote).
8. Horizontal asymptotes of rational functionsCompare degrees: if numerator degree < denominator degree, y = 0. If equal, y = ratio of leading coefficients. If numerator degree is larger, there is no horizontal asymptote.
9. State the epsilon-delta definition of lim as x→a of f(x) = L.For every ε > 0 there is a δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε.
10. Evaluate lim as h→0 of [(2 + h)^3 − 8]/h without expanding.This is the definition of the derivative of x^3 at x = 2. Since d/dx [x^3] = 3x^2, the limit is 3(2)^2 = 12.
11. Equation of the tangent line to y = f(x) at x = ay = f(a) + f'(a)(x − a). Example: for f(x) = x^2 at a = 3, y = 9 + 6(x − 3).
12. Differentiate e^(3x^2).By the chain rule: d/dx [e^(3x^2)] = e^(3x^2)·6x.
13. Use logarithmic differentiation to find the derivative of y = x^x.ln y = x ln x. Differentiate: y'/y = ln x + 1. So y' = x^x(ln x + 1).
14. Derivatives of ln|x| and log_b(x)d/dx [ln|x|] = 1/x for x ≠ 0. d/dx [log_b x] = 1/(x ln b).
15. Derivatives of arcsin x and arctan xd/dx [arcsin x] = 1/sqrt(1 − x^2) for |x| < 1. d/dx [arctan x] = 1/(1 + x^2).
16. State Rolle's Theorem.If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then f'(c) = 0 for some c in (a, b).
17. State the Mean Value Theorem.If f is continuous on [a, b] and differentiable on (a, b), then for some c in (a, b), f'(c) = [f(b) − f(a)]/(b − a): the tangent slope equals the average slope somewhere.
18. What important consequence follows from the Mean Value Theorem about f' = 0?If f'(x) = 0 for all x in an interval, then f is constant there. So two functions with the same derivative on an interval differ by a constant.
19. When can you use L'Hôpital's rule?Only for indeterminate forms 0/0 or ∞/∞. Other forms like 0·∞, ∞ − ∞, 1^∞, 0^0 or ∞^0 must first be rewritten as a quotient (often using logarithms).
20. Evaluate lim as x→∞ of (ln x)/x.It is ∞/∞, so apply L'Hôpital: lim (1/x)/1 = 0. Polynomials grow faster than logarithms.
21. Basic antiderivatives: x^n and 1/x∫ x^n dx = x^(n+1)/(n + 1) + C for n ≠ −1. ∫ (1/x) dx = ln|x| + C. Do not forget the constant of integration.
22. Definite integral as a limit of Riemann sumsThe integral from a to b of f(x) dx = lim as n→∞ of the sum from i = 1 to n of f(x_i*)·Δx, where Δx = (b − a)/n and x_i* is a sample point in the i-th subinterval.
23. Fundamental Theorem of Calculus, Part 1 (with chain rule)If F(x) = integral from a to x of f(t) dt with f continuous, then F'(x) = f(x). If the upper limit is g(x), the derivative is f(g(x))·g'(x).
24. Fundamental Theorem of Calculus, Part 2If F is any antiderivative of a continuous f on [a, b], then the integral from a to b of f(x) dx = F(b) − F(a).
25. Evaluate the integral of sin x from 0 to π.An antiderivative is −cos x, so the integral = −cos π − (−cos 0) = 1 + 1 = 2.
26. Use u-substitution: ∫ 2x cos(x^2) dxLet u = x^2, so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x^2) + C.
27. How do you find the area between two curves?Area = integral from a to b of [top function − bottom function] dx, where a and b are usually the intersection points. If the curves cross, split the interval.
28. Properties: reversing limits and splitting the interval of a definite integralThe integral from b to a of f = −(the integral from a to b of f). The integral from a to c of f = (integral from a to b) + (integral from b to c).
29. Integrals of odd and even functions over [−a, a]If f is odd (f(−x) = −f(x)), the integral from −a to a is 0. If f is even (f(−x) = f(x)), it equals 2 × the integral from 0 to a.
30. What is a differential, and how does it estimate change?dy = f'(x)·dx. For a small change dx in x, the actual change Δy is approximately dy, which is the change along the tangent line.