MATH 125 Final Review: Limits, Derivatives, Applications and the Fundamental Theorem

Terms in this set (31)

1. Informal meaning of lim x->a f(x) = Lf(x) can be made as close to L as we like by taking x close enough to a, but not equal to a. The value f(a) itself does not matter.
2. When does a limit fail to exist?When the left and right limits differ, when f grows without bound, or when f oscillates without settling, as sin(1/x) does near 0.
3. Evaluate lim x->3 (x^2 - 9)/(x - 3)Factor: (x - 3)(x + 3)/(x - 3) = x + 3, so the limit is 6.
4. Key trig limitlim x->0 sin(x)/x = 1. Also lim x->0 (1 - cos x)/x = 0.
5. Three conditions for continuity at x = af(a) is defined, lim x->a f(x) exists, and the limit equals f(a).
6. Intermediate Value TheoremIf 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]. 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. Pitfall: does differentiable imply continuous, and vice versa?Differentiable implies continuous, but continuous does not imply differentiable. |x| is continuous at 0 but has a corner there.
9. Power ruled/dx [x^n] = n*x^(n-1) for any real n. Example: d/dx [x^(1/2)] = (1/2)x^(-1/2).
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). Example: d/dx [sin(x^2)] = cos(x^2) * 2x.
13. 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.
14. Derivatives of e^x and ln xd/dx e^x = e^x; d/dx ln x = 1/x for x > 0. Also d/dx a^x = a^x * ln a.
15. Implicit differentiation: find dy/dx for x^2 + y^2 = 252x + 2y*(dy/dx) = 0, so dy/dx = -x/y.
16. Related rates strategyWrite an equation linking the quantities, differentiate both sides with respect to time t, then substitute known values only after differentiating.
17. Equation of the tangent line at x = ay = f(a) + f'(a)(x - a). This is also the linear approximation of f near a.
18. What is a critical number?A number c in the domain where f'(c) = 0 or f'(c) does not exist. Local extrema can occur only at critical numbers.
19. First derivative testIf f' changes from positive to negative at c, f has a local max there; negative to positive gives a local min; no sign change gives neither.
20. Second derivative test and concavityf'' > 0 means concave up; f'' < 0 means concave down. If f'(c) = 0 and f''(c) > 0, then c is a local min; if f''(c) < 0, a local max.
21. What is an inflection point?A point on the graph where concavity changes, from up to down or down to up.
22. Mean Value TheoremIf f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) - f(a)] / (b - a) for some c in (a, b).
23. Closed interval method for absolute extremaEvaluate f at critical numbers in [a, b] and at endpoints a and b. The largest value is the absolute max; the smallest is the absolute min.
24. L'Hopital's ruleIf a limit gives 0/0 or infinity/infinity, then lim f/g = lim f'/g' when the latter exists. Example: lim x->0 sin x / x = lim cos x / 1 = 1.
25. Antiderivative of x^nIntegral of x^n dx = x^(n+1)/(n+1) + C for n not equal to -1. For n = -1 it is ln|x| + C.
26. Why is + C needed in an indefinite integral?Any two antiderivatives of the same function differ by a constant, so C represents the whole family.
27. Definite integral as a Riemann sumIntegral from a to b of f(x) dx = limit as n -> infinity of the sum of f(x_i)*delta_x, with delta_x = (b - a)/n. It is signed area under the curve.
28. Fundamental Theorem of Calculus, Part 1If F(x) = integral from a to x of f(t) dt with f continuous, then F'(x) = f(x).
29. Fundamental Theorem of Calculus, Part 2Integral from a to b of f(x) dx = F(b) - F(a), where F is any antiderivative of f.
30. Evaluate the integral from 0 to 2 of 3x^2 dxAntiderivative x^3; 2^3 - 0^3 = 8.
31. u-substitution: integral of 2x*cos(x^2) dxLet u = x^2, du = 2x dx, so the integral is integral of cos u du = sin(x^2) + C.