MATH 115 Exam Review: Derivatives, Their Applications, and Introduction to Integration

Terms in this set (30)

1. Average rate of change of f on [a, b][f(b) − f(a)]/(b − a), which is the slope of the secant line through (a, f(a)) and (b, f(b)).
2. Limit definition of the derivative at x = af'(a) = lim as h→0 of [f(a + h) − f(a)]/h, the instantaneous rate of change and the slope of the tangent line at x = a.
3. Interpret f'(5) = 3 if f(t) is gallons of water in a tank at t minutes.At t = 5 minutes, the amount of water is increasing at about 3 gallons per minute. Units of f' are output units per input unit.
4. Exponential growth: compare P = P0·a^t and P = P0·e^(kt), and give the doubling time.Both describe exponential growth when a > 1 or k > 0, with a = e^k. Doubling time = ln 2 / k. If 0 < a < 1 or k < 0, it is exponential decay.
5. Power ruled/dx [x^n] = n·x^(n−1) for any real n. Example: d/dx [x^(1/2)] = (1/2)x^(−1/2).
6. Derivatives of e^x, a^x and ln xd/dx [e^x] = e^x. d/dx [a^x] = (ln a)·a^x. d/dx [ln x] = 1/x for x > 0.
7. 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 = 1/cos^2 x.
8. Product ruled/dx [f·g] = f'·g + f·g'. Example: d/dx [x^2 sin x] = 2x sin x + x^2 cos x.
9. Quotient ruled/dx [f/g] = (f'·g − f·g')/g^2. A common error is reversing the order in the numerator.
10. Chain rule, with an exampled/dx [f(g(x))] = f'(g(x))·g'(x). Example: d/dx [sin(x^2)] = cos(x^2)·2x.
11. Use implicit differentiation to find dy/dx for x^2 + y^2 = 25.Differentiate both sides: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y.
12. Derivative of arctan xd/dx [arctan x] = 1/(1 + x^2). It is found by implicitly differentiating tan y = x.
13. Tangent line (local linear) approximationNear x = a, f(x) ≈ f(a) + f'(a)(x − a). It overestimates when f is concave down and underestimates when f is concave up.
14. Mini-problem: Use a tangent line at x = 4 to approximate sqrt(4.1).f(x) = sqrt(x), f(4) = 2, f'(4) = 1/(2·sqrt 4) = 1/4. So sqrt(4.1) ≈ 2 + (1/4)(0.1) = 2.025.
15. What do the signs of f' and f'' tell you?f' > 0: f increasing; f' < 0: f decreasing. f'' > 0: f concave up (f' increasing); f'' < 0: f concave down.
16. What is a critical point of f?A point in the domain where f'(x) = 0 or f'(x) does not exist. Local extrema of a function can only occur at critical points (or endpoints).
17. First derivative testAt a critical point, if f' changes from positive to negative, f has a local maximum; from negative to positive, a local minimum; if the sign does not change, neither.
18. 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.
19. What is an inflection point?A point on the graph where concavity changes (f'' changes sign). f''(x) = 0 alone is not enough: x^4 has f''(0) = 0 but no inflection point.
20. How do you find the global maximum and minimum of a continuous f on a closed interval [a, b]?Evaluate f at every critical point inside the interval and at both endpoints; the largest value is the global max and the smallest is the global min (guaranteed by the Extreme Value Theorem).
21. Mini-problem: A rectangle has perimeter 40. What dimensions maximize its area?With sides x and 20 − x, A = x(20 − x). A'(x) = 20 − 2x = 0 gives x = 10, so a 10 by 10 square with area 100.
22. Mini-problem: A circle's radius grows at 2 cm/s. How fast is area changing when r = 5 cm?A = πr^2, so dA/dt = 2πr·(dr/dt) = 2π(5)(2) = 20π ≈ 62.8 cm^2/s.
23. Marginal analysis: when is profit maximized?Profit π = R − C is maximized where π' = 0, that is, where marginal revenue equals marginal cost (R'(q) = C'(q)), with MC rising through MR.
24. State L'Hôpital's rule.If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), if the latter exists. Example: lim as x→0 of sin x / x = lim cos x / 1 = 1.
25. Continuity vs differentiabilityIf f is differentiable at a, it is continuous at a. The converse fails: |x| is continuous at 0 but not differentiable there because of the corner.
26. Left vs right Riemann sums for an increasing functionFor an increasing function, the left sum underestimates and the right sum overestimates the definite integral. For a decreasing function, the reverse is true.
27. What does the definite integral of f from a to b represent?The limit of Riemann sums as the number of subintervals goes to infinity: the net signed area between the graph and the x-axis (area above counts positive, area below counts negative).
28. State the Fundamental Theorem of Calculus as total change.If F' is continuous on [a, b], then the integral from a to b of F'(t) dt = F(b) − F(a): integrating a rate of change gives the total change.
29. If v(t) is velocity, what do the integral of v(t) and the integral of |v(t)| give?The integral of v(t) from a to b gives displacement (net change in position). The integral of |v(t)| gives total distance traveled.
30. Average value of f on [a, b]Average value = (1/(b − a)) × integral from a to b of f(x) dx.