MATH 140 Midterm Review: Functions, Limits, Continuity and Derivatives

Terms in this set (30)

1. What is the domain of a composite function f(g(x))?All x in the domain of g for which g(x) is in the domain of f. For f(x) = sqrt(x) and g(x) = x - 2, the domain of f(g(x)) is x >= 2.
2. When does a function have an inverse, and how do you find it?When it is one-to-one (passes the horizontal line test). Swap x and y and solve for y. The graph of the inverse is the reflection across y = x.
3. When does a two-sided limit exist?lim as x -> a of f(x) = L exists only if both one-sided limits exist and are equal to L. The function's value at a does not matter.
4. Mini-problem: lim as x -> 3 of (x^2 - 9)/(x - 3)Factor: (x - 3)(x + 3)/(x - 3) = x + 3 for x not equal to 3, so the limit is 6.
5. Mini-problem: lim as x -> 0 of (sqrt(x + 4) - 2)/xMultiply by the conjugate: x / (x(sqrt(x + 4) + 2)) = 1/(sqrt(x + 4) + 2). As x -> 0 this gives 1/4.
6. Squeeze theoremIf g(x) <= f(x) <= h(x) near a and lim g = lim h = L, then lim f = L. Example: -x^2 <= x^2 sin(1/x) <= x^2, so x^2 sin(1/x) -> 0 as x -> 0.
7. Important trigonometric limitlim as x -> 0 of sin(x)/x = 1 (x in radians). Also lim as x -> 0 of (1 - cos x)/x = 0.
8. Limits at infinity of rational functionsCompare degrees. Numerator degree lower: limit 0. Equal degrees: ratio of leading coefficients. Numerator degree higher: no finite limit (grows without bound).
9. Mini-problem: lim as x -> infinity of (3x^2 + 1)/(5x^2 - x)Equal degrees, so the limit is the ratio of leading coefficients: 3/5.
10. What is a vertical asymptote?x = a is a vertical asymptote if f(x) approaches positive or negative infinity as x approaches a from at least one side, often where a denominator is zero but the numerator is not.
11. Definition of continuity at a pointf is continuous at a if f(a) is defined, lim as x -> a of f(x) exists, and the limit equals f(a).
12. Removable vs jump vs infinite discontinuityRemovable: the limit exists but differs from f(a) or f(a) is undefined (a hole). Jump: the one-sided limits exist but differ. Infinite: the function goes to infinity near the point.
13. Intermediate Value TheoremIf f is continuous on [a, b] and N is between f(a) and f(b), then some c in (a, b) has f(c) = N. Often used to show an equation has a root.
14. Limit definition of the derivativef'(x) = lim as h -> 0 of (f(x + h) - f(x)) / h. It gives the slope of the tangent line and the instantaneous rate of change.
15. Does continuity imply differentiability?No. Differentiability implies continuity, but not the reverse. f(x) = |x| is continuous at 0 but not differentiable there because of the corner.
16. Power ruled/dx [x^n] = n·x^(n-1) for any real n. For example, d/dx [x^(-2)] = -2x^(-3).
17. Product ruled/dx [f·g] = f'·g + f·g'.
18. Quotient ruled/dx [f/g] = (f'·g - f·g') / g^2. Order matters in the numerator: derivative of the top times the bottom comes first.
19. Chain rule with an exampled/dx [f(g(x))] = f'(g(x))·g'(x). Example: d/dx [sin(3x^2)] = cos(3x^2)·6x.
20. Derivatives of the six trig functionssin -> cos; cos -> -sin; tan -> sec^2; cot -> -csc^2; sec -> sec·tan; csc -> -csc·cot.
21. Derivatives of e^x and ln xd/dx [e^x] = e^x. d/dx [ln x] = 1/x for x > 0. With the chain rule, d/dx [e^(g(x))] = e^(g(x))·g'(x).
22. Mini-problem: equation of the tangent line to y = x^2 at x = 3Point (3, 9), slope 2(3) = 6. Tangent line: y - 9 = 6(x - 3), so y = 6x - 9.
23. Logarithmic differentiation of y = x^xln y = x ln x. Differentiate: y'/y = ln x + 1. So y' = x^x (ln x + 1).
24. Implicit differentiation: key ideaDifferentiate both sides with respect to x, treating y as a function of x, so every y term gets a factor of dy/dx by the chain rule. Then solve for dy/dx.
25. Position, velocity and accelerationIf s(t) is position, velocity is v(t) = s'(t) and acceleration is a(t) = v'(t) = s''(t). The object speeds up when v and a have the same sign.
26. What is an antiderivative?F is an antiderivative of f if F'(x) = f(x). All antiderivatives differ by a constant, so the general antiderivative of x^n (n not -1) is x^(n+1)/(n+1) + C.
27. Mini-problem: f'(x) = 6x and f(0) = 2. Find f(x).General antiderivative: f(x) = 3x^2 + C. Using f(0) = 2 gives C = 2, so f(x) = 3x^2 + 2.
28. Using f' to find where f is increasing or decreasingf is increasing where f'(x) > 0 and decreasing where f'(x) < 0. Find critical points, then test the sign of f' on each interval.
29. Using f'' to find concavityf is concave up where f''(x) > 0 and concave down where f''(x) < 0. A point where concavity changes is an inflection point.
30. Common pitfall: derivative of a productThe derivative of f·g is not f'·g'. Use the product rule f'g + fg'. Similarly, (f/g)' is not f'/g'.

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