MATH-UA 121 Exam Review: Limits, Derivatives, Applications and Intro Integrals

Terms in this set (30)

1. Formal idea of lim x->a f(x) = Lf(x) can be made as close as you like to L by taking x sufficiently close to a (but not equal to a). The value f(a) itself does not matter.
2. When does a two-sided limit exist?Only when both one-sided limits exist and are equal: lim x->a- f(x) = lim x->a+ f(x).
3. Evaluate lim x->2 (x^2 - 4)/(x - 2)Factor: (x - 2)(x + 2)/(x - 2) = x + 2 for x not equal to 2, so the limit is 4.
4. lim x->0 sin(x)/x1 (with x in radians). It is the key limit used to derive d/dx [sin x] = cos x.
5. Three conditions for f to be continuous at x = af(a) is defined, lim x->a f(x) exists, and lim x->a f(x) = f(a).
6. Intermediate Value TheoremIf f is continuous on [a, b] and N is between f(a) and f(b), then there is some c in (a, b) with f(c) = N. Often used to show a root exists.
7. Limit definition of the derivativef'(x) = lim h->0 [f(x + h) - f(x)] / h, the slope of the tangent line, or instantaneous rate of change.
8. Differentiable vs continuousDifferentiability at a point implies continuity there, but not the reverse. Example: f(x) = |x| is continuous at 0 but not differentiable (sharp corner).
9. Power ruled/dx [x^n] = n*x^(n-1) for any real constant n.
10. Product ruled/dx [f*g] = f'*g + f*g'.
11. Quotient ruled/dx [f/g] = (f'*g - f*g') / g^2.
12. Chain ruled/dx [f(g(x))] = f'(g(x)) * g'(x). Differentiate the outer function at the inner one, then multiply by the derivative of the inner function.
13. Differentiate y = (3x^2 + 1)^5By the chain rule, y' = 5(3x^2 + 1)^4 * 6x = 30x(3x^2 + 1)^4.
14. Derivatives of sin x, cos x and tan xd/dx sin x = cos x; d/dx cos x = -sin x; d/dx tan x = sec^2 x.
15. Derivatives of e^x and ln xd/dx e^x = e^x; d/dx ln x = 1/x for x > 0. More generally d/dx a^x = a^x * ln a.
16. Implicit differentiation: find dy/dx for x^2 + y^2 = 25Differentiate both sides: 2x + 2y*(dy/dx) = 0, so dy/dx = -x/y.
17. Strategy for related rates problemsDraw and label a diagram, write an equation relating the quantities, differentiate both sides with respect to time t, then substitute known values only after differentiating.
18. Linear approximation formulaf(x) is approximately f(a) + f'(a)(x - a) for x near a; this is the tangent line at x = a.
19. Critical number definitionA number c in the domain of f where f'(c) = 0 or f'(c) does not exist. Local extrema can only occur at critical numbers.
20. First derivative testIf f' changes from positive to negative at c, f has a local max at c; from negative to positive, a local min; no sign change means no local extremum.
21. Second derivative testIf f'(c) = 0 and f''(c) > 0, local min at c; if f''(c) < 0, local max; if f''(c) = 0 the test is inconclusive.
22. Concavity and inflection pointsf'' > 0 means concave up and f'' < 0 means concave down. An inflection point is where concavity changes.
23. Closed interval method for absolute extremaFor continuous f on [a, b], evaluate f at all critical numbers in (a, b) and at the endpoints a and b; the largest value is the absolute max and the smallest is the absolute min.
24. Mean Value TheoremIf f is continuous on [a, b] and differentiable on (a, b), there is some c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a).
25. L'Hopital's ruleIf lim f/g gives 0/0 or infinity/infinity, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists. Do not use it for other forms without rewriting first.
26. Horizontal asymptote of (3x^2 + 1)/(5x^2 - x)Degrees of numerator and denominator are equal, so the limit as x goes to infinity is the ratio of leading coefficients: y = 3/5.
27. Antiderivative of x^n (n not -1)Integral of x^n dx = x^(n+1)/(n+1) + C. For n = -1, the integral of 1/x dx = ln|x| + C.
28. Fundamental Theorem of Calculus, Part 2If F is an antiderivative of continuous f on [a, b], then the integral from a to b of f(x) dx = F(b) - F(a).
29. Compute the integral from 0 to 2 of 3x^2 dxAn antiderivative is x^3, so the value is 2^3 - 0^3 = 8.
30. Fundamental Theorem of Calculus, Part 1If g(x) = integral from a to x of f(t) dt with f continuous, then g'(x) = f(x).