Calculus 1000 Exam Review: Derivatives, L'Hospital's Rule, Definite Integrals and Applications

Terms in this set (29)

1. Evaluate lim (x -> infinity) (3x^2 + 1)/(5x^2 - x).Divide by x^2: (3 + 1/x^2)/(5 - 1/x) -> 3/5.
2. Evaluate lim (x -> 0) (e^x - 1)/x.1. It is 0/0, and L'Hospital's Rule gives lim e^x/1 = 1.
3. Derivative of ln(x^2 + 1)2x/(x^2 + 1), by the chain rule.
4. Derivative of e^(3x) · x^2Product rule: 3e^(3x)·x^2 + 2x·e^(3x) = x·e^(3x)(3x + 2).
5. Derivative of log base b of x1/(x ln b).
6. Derivatives of sin x, cos x and sec xd/dx sin x = cos x; d/dx cos x = -sin x; d/dx sec x = sec x tan x.
7. Derivative of arccos x-1/sqrt(1 - x^2), for -1 < x < 1.
8. What are the domain and range of arcsin x and arctan x?arcsin: domain [-1, 1], range [-pi/2, pi/2]. arctan: domain all reals, range (-pi/2, pi/2).
9. Evaluate arcsin(1/2) and arctan(1).arcsin(1/2) = pi/6; arctan(1) = pi/4.
10. Mini-problem: differentiate y = arctan(2x).y' = 2/(1 + 4x^2).
11. Which indeterminate forms can L'Hospital's Rule be applied to directly?Only 0/0 and infinity/infinity. Other forms (0·infinity, infinity - infinity, 1^infinity, 0^0, infinity^0) must first be rewritten as a quotient.
12. How do you evaluate a 1^infinity limit like lim (x -> infinity) (1 + 1/x)^x?Let y = (1 + 1/x)^x, so ln y = x ln(1 + 1/x) = ln(1 + 1/x)/(1/x), a 0/0 form that tends to 1. So y -> e^1 = e.
13. Evaluate lim (x -> 0+) x ln x.Rewrite as ln x/(1/x) (infinity/infinity form). L'Hospital: (1/x)/(-1/x^2) = -x -> 0.
14. Common L'Hospital pitfallApplying it when the limit is not indeterminate, or using the quotient rule on f/g instead of differentiating numerator and denominator separately.
15. Riemann sum definition of the definite integralIntegral from a to b of f(x) dx = lim (n -> infinity) of sum from i = 1 to n of f(x_i*)·delta x, where delta x = (b - a)/n.
16. 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).
17. Mini-problem: find d/dx of the integral from 0 to x^2 of cos(t) dt.By FTC Part 1 and the chain rule: cos(x^2)·2x.
18. 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).
19. Evaluate the integral from 0 to pi of sin x dx.[-cos x] from 0 to pi = -cos(pi) + cos(0) = 1 + 1 = 2.
20. Basic antiderivatives: x^n, 1/x, e^xx^(n+1)/(n+1) + C for n not equal to -1; ln|x| + C; e^x + C.
21. Antiderivatives of 1/(1 + x^2) and 1/sqrt(1 - x^2)arctan x + C and arcsin x + C.
22. How does u-substitution work? Integrate 2x·cos(x^2) dx.Let u = x^2, du = 2x dx, so the integral becomes integral of cos u du = sin u + C = sin(x^2) + C.
23. When changing variables in a definite integral, what must you do with the limits?Either convert the limits to u-values (u(a) and u(b)) or substitute back to x before evaluating with the original limits. Do not mix the two.
24. Area between two curvesArea = integral from a to b of [top function - bottom function] dx, where a and b are often the intersection points; split the integral if the curves cross.
25. Mini-problem: area between y = x and y = x^2 on [0, 1]Integral from 0 to 1 of (x - x^2) dx = 1/2 - 1/3 = 1/6.
26. Disk method formula for volume of revolution about the x-axisV = pi · integral from a to b of [f(x)]^2 dx.
27. Washer method formulaV = pi · integral from a to b of ([R(x)]^2 - [r(x)]^2) dx, where R is the outer radius and r the inner radius.
28. Mini-problem: volume when y = sqrt(x) on [0, 4] is rotated about the x-axisV = pi · integral from 0 to 4 of x dx = pi · (16/2) = 8·pi.
29. Does the integral of f from a to b always equal area?No. It gives net signed area: regions below the x-axis count as negative. Use |f(x)| to get total area.