MAT 1320 Final Review: Applications of Derivatives, Integration Techniques and Numerical Methods

Terms in this set (30)

1. State L'Hospital's ruleIf lim f(x)/g(x) gives 0/0 or infinity/infinity, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the second limit exists. Differentiate top and bottom separately, not as a quotient.
2. Mini-problem: lim as x -> 0 of (e^x - 1)/xIt is 0/0, so apply L'Hospital: lim e^x / 1 = e^0 = 1.
3. Mini-problem: lim as x -> infinity of x/e^xIt is infinity/infinity. L'Hospital gives lim 1/e^x = 0.
4. How do you handle the indeterminate form 0 x infinity?Rewrite the product as a quotient, for example f·g = f / (1/g), to get 0/0 or infinity/infinity, then apply L'Hospital's rule.
5. 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): the instantaneous rate equals the average rate somewhere.
6. Extreme Value TheoremA function continuous on a closed interval [a, b] attains both an absolute maximum and an absolute minimum there. Check critical points and endpoints to find them.
7. Critical point and the first derivative testA critical point is where f'(c) = 0 or f'(c) does not exist. If f' changes from + to - at c, f has a local max; from - to +, a local min; no sign change, neither.
8. Second derivative test and inflection pointsIf f'(c) = 0 and f''(c) > 0, local min; if f''(c) < 0, local max; if f''(c) = 0, the test is inconclusive. An inflection point is where concavity changes sign.
9. Linear approximation formula and an exampleL(x) = f(a) + f'(a)(x - a). For sqrt(4.1) with a = 4: f(4) = 2 and f'(4) = 1/(2·2) = 1/4, so sqrt(4.1) is about 2 + 0.1/4 = 2.025.
10. Newton's method formulax(n+1) = x(n) - f(x(n)) / f'(x(n)). It repeatedly uses tangent lines to approximate a root and can fail if f'(x(n)) is near zero or the starting guess is poor.
11. Mini-problem: a circle's radius grows at 2 cm/s. How fast is area growing when r = 5 cm?A = pi r^2, so dA/dt = 2 pi r (dr/dt) = 2 pi (5)(2) = 20 pi cm^2/s, about 62.8 cm^2/s.
12. Mini-problem: find dy/dx for x^2 + y^2 = 25Differentiate implicitly: 2x + 2y (dy/dx) = 0, so dy/dx = -x/y.
13. Derivatives of ln x, arctan x and arcsin xd/dx [ln x] = 1/x. d/dx [arctan x] = 1/(1 + x^2). d/dx [arcsin x] = 1/sqrt(1 - x^2).
14. Derivatives of a^x and log base a of xd/dx [a^x] = a^x ln a. d/dx [log_a x] = 1/(x ln a).
15. Definite integral as a limit of Riemann sumsThe integral from a to b of f(x) dx = lim as n -> infinity of the sum of f(x_i*) delta x, where delta x = (b - a)/n. It gives net signed area.
16. Fundamental Theorem of Calculus, Part 1If F(x) = integral from a to x of f(t) dt, then F'(x) = f(x). With a variable upper limit g(x), the derivative is f(g(x))·g'(x).
17. Fundamental Theorem of Calculus, Part 2If F is any antiderivative of a continuous f, then the integral from a to b of f(x) dx = F(b) - F(a).
18. Mini-problem: integral from 0 to 2 of 3x^2 dxAntiderivative x^3, so 2^3 - 0^3 = 8.
19. Mini-problem: integral of 2x cos(x^2) dxLet u = x^2, du = 2x dx. The integral becomes integral of cos u du = sin(x^2) + C.
20. Integration by parts formula and a tip for choosing uIntegral of u dv = uv - integral of v du. Choose u in the order LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
21. Mini-problem: integral of x e^x dxLet u = x and dv = e^x dx, so du = dx and v = e^x. Result: x e^x - e^x + C.
22. Mini-problem: integral of ln x dxLet u = ln x and dv = dx, so du = dx/x and v = x. Result: x ln x - x + C.
23. Mini-problem: integral of 1/(x(x + 1)) dxPartial fractions: 1/(x(x+1)) = 1/x - 1/(x+1). Result: ln|x| - ln|x + 1| + C.
24. Antiderivatives of 1/x and 1/(1 + x^2)Integral of 1/x dx = ln|x| + C. Integral of 1/(1 + x^2) dx = arctan x + C.
25. Trapezoidal ruleIntegral from a to b of f is about (delta x / 2)[f(x0) + 2f(x1) + 2f(x2) + ... + 2f(x(n-1)) + f(xn)], with delta x = (b - a)/n.
26. Simpson's ruleRequires n even: integral is about (delta x / 3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 4f(x(n-1)) + f(xn)]. It fits parabolas and is exact for cubics.
27. Error bounds for midpoint, trapezoidal and Simpson's rulesIf |f''| <= K: midpoint error <= K(b-a)^3/(24n^2), trapezoid error <= K(b-a)^3/(12n^2). If |f''''| <= K: Simpson error <= K(b-a)^5/(180n^4).
28. Forward vs central difference approximations of f'(a)Forward: (f(a + h) - f(a)) / h, error proportional to h. Central: (f(a + h) - f(a - h)) / (2h), error proportional to h^2, so it is usually more accurate.
29. Mini-problem: a rectangle has perimeter 40. What dimensions maximize area?Area A = x(20 - x). A'(x) = 20 - 2x = 0 gives x = 10, so a 10 by 10 square with area 100.
30. Common pitfall with u-substitution in definite integralsWhen you substitute, either change the limits to u-values or convert back to x before evaluating. Mixing new integrand with old limits gives wrong answers.