MATH 151 Exam Review: Vectors, Limits, Derivatives and Integration

Terms in this set (30)

1. Dot product of a = <a1, a2> and b = <b1, b2>, and its geometric meaninga . b = a1*b1 + a2*b2 = |a| |b| cos(theta), where theta is the angle between them. A dot product of 0 means the vectors are orthogonal.
2. Find the angle between <1, 0> and <1, 1>.cos(theta) = 1 / (1 * sqrt(2)), so theta = pi/4 (45 degrees).
3. Formula for the scalar projection of b onto acomp_a b = (a . b) / |a|. The vector projection is ((a . b) / |a|^2) a.
4. Parametric equation of the line through point P0 in direction vr(t) = P0 + t*v, i.e. x = x0 + t*v1, y = y0 + t*v2.
5. What does lim x->a f(x) = L mean informally?f(x) gets arbitrarily close to L as x gets close to a (from both sides), regardless of the value of f at a.
6. Evaluate lim x->3 (x^2 - 9)/(x - 3).Factor: (x - 3)(x + 3)/(x - 3) = x + 3, so the limit is 6.
7. What are the two key special limits involving sin x?lim x->0 sin(x)/x = 1 and lim x->0 (1 - cos x)/x = 0.
8. 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).
9. State the Intermediate Value Theorem.If 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.
10. Limit definition of the derivativef'(x) = lim h->0 [f(x + h) - f(x)] / h, the slope of the tangent line at x.
11. How do you find horizontal asymptotes of a rational function?Compare degrees: if the numerator degree is lower, y = 0; if equal, y = ratio of leading coefficients; if higher, there is no horizontal asymptote.
12. Power rule, product rule and quotient ruled/dx [x^n] = n*x^(n-1). (fg)' = f'g + fg'. (f/g)' = (f'g - fg') / g^2.
13. State the chain rule and differentiate sin(3x^2).d/dx f(g(x)) = f'(g(x)) * g'(x). So d/dx sin(3x^2) = cos(3x^2) * 6x.
14. Derivatives of sin x, cos x, tan x, e^x and ln xcos x, -sin x, sec^2 x, e^x, and 1/x.
15. Find dy/dx if x^2 + y^2 = 25.Implicit differentiation: 2x + 2y dy/dx = 0, so dy/dx = -x/y.
16. Velocity and acceleration from a position function s(t)Velocity v(t) = s'(t); acceleration a(t) = v'(t) = s''(t). For a vector function r(t), speed is |r'(t)|.
17. Equation of the tangent line to y = f(x) at x = ay = f(a) + f'(a)(x - a).
18. Approximate sqrt(4.1) using linearization.With f(x) = sqrt(x) at a = 4: L(x) = 2 + (1/4)(x - 4), so sqrt(4.1) is approximately 2 + 0.025 = 2.025.
19. General strategy for related rates problemsDraw and label, write an equation relating the variables, differentiate both sides with respect to t, then substitute known values only after differentiating.
20. How do you find absolute extrema of a continuous f on [a, b]?Evaluate f at critical points (where f' = 0 or is undefined) inside the interval and at the endpoints; the largest and smallest values are the absolute max and min.
21. First vs second derivative testFirst: a critical point is a max if f' changes + to -, a min if - to +. Second: if f'(c) = 0 and f''(c) > 0 it is a local min; if f''(c) < 0 a local max; f''(c) = 0 is inconclusive.
22. State the Mean Value Theorem.If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a).
23. When can you use L'Hopital's rule?For limits of the indeterminate form 0/0 or infinity/infinity: lim f/g = lim f'/g', provided the new limit exists.
24. What does an inflection point mean?A point where f is continuous and concavity changes (f'' changes sign).
25. Antiderivative of x^n (n not -1) and of 1/xIntegral of x^n dx = x^(n+1)/(n+1) + C. Integral of 1/x dx = ln|x| + C.
26. State the Fundamental Theorem of Calculus (both parts).Part 1: if F(x) = integral from a to x of f(t) dt, then F'(x) = f(x). Part 2: integral from a to b of f(x) dx = F(b) - F(a), where F is any antiderivative of f.
27. Evaluate the integral from 0 to 2 of 3x^2 dx.Antiderivative x^3, so 2^3 - 0^3 = 8.
28. Integrate 2x*cos(x^2) dx using substitution.Let u = x^2, du = 2x dx, so the integral is sin(u) + C = sin(x^2) + C.
29. What does a Riemann sum approximate?The definite integral: the sum of f(x_i*) * delta x over n subintervals approximates the net signed area under f, becoming exact as n -> infinity.
30. Common pitfall: is d/dx [e^(2x)] equal to e^(2x)?No. By the chain rule it is 2*e^(2x).