MATH 221 Exam Review: Limits, Derivatives, Applications and Integrals

Terms in this set (30)

1. Formal idea 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-hand and right-hand limits differ, when the function grows without bound, or when it oscillates without settling down.
3. lim x->0 of sin(x)/x1 (with x in radians).
4. Evaluate lim x->2 of (x^2 - 4)/(x - 2)Factor: (x - 2)(x + 2)/(x - 2) = x + 2, so the limit is 4.
5. 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).
6. Intermediate Value TheoremIf f is continuous on [a, b] and N is between f(a) and f(b), then there is a 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 of [f(x + h) - f(x)] / h. It gives the slope of the tangent line and the instantaneous rate of change.
8. Does differentiable imply continuous? Does continuous imply differentiable?Differentiable implies continuous. Continuous does not imply differentiable; f(x) = |x| is continuous at 0 but has a corner there.
9. Power ruled/dx [x^n] = n*x^(n-1) for any real n.
10. Product rule and quotient rule(fg)' = f'g + fg'. (f/g)' = (f'g - fg') / g^2.
11. Chain ruled/dx [f(g(x))] = f'(g(x)) * g'(x). Example: d/dx [sin(x^2)] = cos(x^2) * 2x.
12. 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.
13. Derivatives of e^x, a^x and ln xd/dx e^x = e^x, d/dx a^x = a^x * ln a, d/dx ln x = 1/x for x > 0.
14. Derivatives of arcsin x and arctan xd/dx arcsin x = 1/sqrt(1 - x^2). d/dx arctan x = 1/(1 + x^2).
15. Implicit differentiation: find dy/dx for x^2 + y^2 = 252x + 2y*(dy/dx) = 0, so dy/dx = -x/y.
16. Steps for a related rates problemDraw and label variables, write an equation relating them, differentiate both sides with respect to time t, then substitute the known values only after differentiating.
17. 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.
18. Mean Value TheoremIf f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f'(c) = [f(b) - f(a)] / (b - a).
19. First derivative test for local extremaAt a critical point, if f' changes from + to - there is a local max; from - to + a local min; no sign change means neither.
20. Second derivative: concavity and the second derivative testf'' > 0 means concave up, f'' < 0 means concave down. At a critical point c, f''(c) > 0 gives a local min and f''(c) < 0 gives a local max; f''(c) = 0 is inconclusive.
21. How to find absolute extrema on a closed interval [a, b]Evaluate f at every critical point in (a, b) and at both endpoints. The largest value is the absolute max and the smallest is the absolute min.
22. L'Hopital's ruleIf lim f/g gives 0/0 or infinity/infinity, then lim f/g = lim f'/g' (if that limit exists). Differentiate top and bottom separately, not as a quotient.
23. Fundamental Theorem of Calculus, Part 1If F(x) = integral from a to x of f(t) dt and f is continuous, then F'(x) = f(x).
24. Fundamental Theorem of Calculus, Part 2If F is any antiderivative of continuous f, then the integral from a to b of f(x) dx = F(b) - F(a).
25. Antiderivative of x^n and of 1/xIntegral of x^n dx = x^(n+1)/(n+1) + C for n not equal to -1. Integral of 1/x dx = ln|x| + C.
26. u-substitution: integrate 2x*cos(x^2) dxLet u = x^2, du = 2x dx. Integral of cos u du = sin u + C = sin(x^2) + C.
27. Riemann sum and the definite integralSum of f(x_i)*delta x over n subintervals. As n goes to infinity the sum approaches the definite integral, the signed area under the curve.
28. Average value of f on [a, b]f_avg = [1/(b - a)] * integral from a to b of f(x) dx.
29. Area between curves y = f(x) and y = g(x) where f >= g on [a, b]Area = integral from a to b of [f(x) - g(x)] dx.
30. Solution of the differential equation dy/dt = k*yy(t) = y0 * e^(k*t). k > 0 gives exponential growth, k < 0 gives decay.