MATH 100 Final Review: Limits, Derivative Rules, Optimization and Approximation

Terms in this set (31)

1. Limit definition of the derivativef'(a) = lim as h→0 of [f(a + h) − f(a)] / h, the slope of the tangent line at x = a and the instantaneous rate of change.
2. Three conditions for f to be continuous at x = af(a) is defined, the limit of f(x) as x→a exists, and that limit equals f(a).
3. Does continuity imply differentiability?No. Differentiable implies continuous, but not the reverse. Example: f(x) = |x| is continuous at 0 but has a corner there, so f'(0) does not exist.
4. Power ruled/dx [x^n] = n·x^(n−1) for any real n. Example: d/dx [x^(1/2)] = (1/2)x^(−1/2).
5. Product ruled/dx [f·g] = f'·g + f·g'.
6. 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.
7. Chain ruled/dx [f(g(x))] = f'(g(x))·g'(x). Example: d/dx [(3x + 1)^5] = 5(3x + 1)^4·3 = 15(3x + 1)^4.
8. 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. These hold only when x is in radians.
9. 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.
10. Derivatives of arcsin x and arctan xd/dx arcsin x = 1 / √(1 − x^2); d/dx arctan x = 1 / (1 + x^2).
11. Evaluate lim as x→0 of sin(x)/x.The limit is 1 (with x in radians). It is the key limit used to prove that d/dx sin x = cos x.
12. Use implicit differentiation on x^2 + y^2 = 25 to find dy/dx.Differentiate both sides: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y. At (3, 4) the slope is −3/4.
13. Differentiate y = x^x.Take logs: ln y = x·ln x. Differentiate: y'/y = ln x + 1, so y' = x^x·(ln x + 1).
14. Equation of the tangent line to y = f(x) at x = ay = f(a) + f'(a)(x − a).
15. Use a linear approximation to estimate √4.1.With f(x) = √x at a = 4: f(4) = 2 and f'(4) = 1/4. So √4.1 ≈ 2 + (1/4)(0.1) = 2.025 (actual ≈ 2.0248).
16. Related rates: how is dV/dt related to dr/dt for a sphere?V = (4/3)πr^3, so dV/dt = 4πr^2·(dr/dt). Differentiate with respect to time before substituting the values at the given instant.
17. What is a critical point?A point c in the domain of f where f'(c) = 0 or f'(c) does not exist. Local extrema can only occur at critical points (or endpoints).
18. First derivative testIf f' changes from + to − at a critical point, it is a local maximum; from − to +, a local minimum; if f' does not change sign, it is neither.
19. Second derivative testIf f'(c) = 0 and f''(c) > 0, f has a local minimum at c; if f''(c) < 0, a local maximum. If f''(c) = 0 the test is inconclusive.
20. What is an inflection point?A point on the graph where the concavity changes (f'' changes sign). f''(c) = 0 alone is not enough: f(x) = x^4 has f''(0) = 0 but no inflection point.
21. How do you find absolute extrema of a continuous f on a closed interval [a, b]?Evaluate f at every critical point inside (a, b) and at both endpoints; the largest value is the absolute max and the smallest is the absolute min (guaranteed by the Extreme Value Theorem).
22. State the Mean Value Theorem.If 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).
23. State the Intermediate Value Theorem.If f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for some c in [a, b]. It is often used to show an equation has a root.
24. When can L'Hôpital's rule be used?For limits of the indeterminate form 0/0 or ∞/∞: then lim f/g = lim f'/g', if the latter exists. Other forms like 0·∞ must first be rewritten as a quotient.
25. Solve dy/dt = ky and give the doubling time.y(t) = y0·e^(kt), where y0 = y(0). For k > 0 the doubling time is ln 2 / k.
26. Newton's method formulax_(n+1) = x_n − f(x_n) / f'(x_n). Each step follows the tangent line to where it hits the x-axis. It fails if f'(x_n) = 0.
27. Taylor polynomial of degree n for f about x = aT_n(x) = sum for k = 0 to n of f^(k)(a)/k! · (x − a)^k. Example: about 0, e^x ≈ 1 + x + x^2/2 + x^3/6.
28. Find lim as x→∞ of (3x^2 + 1)/(5x^2 − x).Divide top and bottom by x^2: the limit is 3/5. When the degrees are equal, the limit is the ratio of leading coefficients, giving the horizontal asymptote y = 3/5.
29. Use the squeeze theorem to find lim as x→0 of x^2·sin(1/x).Since −x^2 ≤ x^2·sin(1/x) ≤ x^2 and both bounds → 0, the limit is 0.
30. Optimization: a rectangle has perimeter 40. What dimensions maximize its area?With sides x and 20 − x, A = x(20 − x). A' = 20 − 2x = 0 gives x = 10. A 10 by 10 square gives the maximum area of 100.
31. Common pitfall: what is d/dx [e^(x^2)]?By the chain rule it is 2x·e^(x^2). It is NOT x^2·e^(x^2 − 1); the power rule does not apply when the variable is in the exponent.