MATH 140 Exam Review: Limits, Derivative Rules, Related Rates and Optimization

Terms in this set (30)

1. Definition of the derivative as a limitf'(x) = lim as h -> 0 of [f(x+h) - f(x)] / h, the instantaneous rate of change and the slope of the tangent line.
2. Evaluate lim as x -> 3 of (x^2 - 9)/(x - 3).Factor: (x-3)(x+3)/(x-3) = x + 3, so the limit is 6.
3. lim as x -> 0 of sin(x)/xEquals 1 (x in radians). This is the key limit used to derive d/dx [sin x] = cos x.
4. Three conditions for f to be continuous at x = af(a) is defined, lim as x -> a of f(x) exists, and the limit equals f(a).
5. State the Intermediate Value Theorem.If 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 a root exists.
6. How do you find horizontal asymptotes of a rational function?Compare degrees: numerator degree lower gives y = 0; equal degrees gives y = ratio of leading coefficients; numerator higher gives none.
7. Power ruled/dx [x^n] = n*x^(n-1) for any real n. Example: d/dx [x^5] = 5x^4.
8. Product ruled/dx [f*g] = f'*g + f*g'.
9. Quotient ruled/dx [f/g] = (f'*g - f*g') / g^2. Mnemonic: low d-high minus high d-low, over low squared.
10. Chain ruled/dx [f(g(x))] = f'(g(x)) * g'(x). Example: d/dx [(3x+1)^4] = 4(3x+1)^3 * 3 = 12(3x+1)^3.
11. 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.
12. Derivatives of e^x and ln xd/dx [e^x] = e^x; d/dx [ln x] = 1/x for x > 0. More generally d/dx [a^x] = a^x * ln a.
13. Differentiate y = e^(x^2).By the chain rule, y' = e^(x^2) * 2x = 2x*e^(x^2).
14. Find the tangent line to y = x^2 at x = 3.y' = 2x = 6 at x = 3 and y(3) = 9, so the tangent is y - 9 = 6(x - 3), i.e. y = 6x - 9.
15. What is implicit differentiation? Find dy/dx for x^2 + y^2 = 25.Differentiate both sides treating y as a function of x: 2x + 2y*dy/dx = 0, so dy/dx = -x/y.
16. Derivatives of inverse trig: arcsin x and arctan xd/dx [arcsin x] = 1/sqrt(1 - x^2); d/dx [arctan x] = 1/(1 + x^2).
17. Steps for a related rates problemDraw and label, write an equation relating the quantities, differentiate with respect to time t, then substitute known values only after differentiating.
18. A circle's radius grows at 2 cm/s. How fast is its area growing when r = 5 cm?A = pi*r^2, so dA/dt = 2*pi*r*dr/dt = 2*pi*5*2 = 20*pi cm^2/s.
19. What is a critical number?A value c in the domain of f where f'(c) = 0 or f'(c) does not exist. Local extrema can only occur there.
20. First derivative testIf f' changes from positive to negative at c, f has a local max; from negative to positive, a local min; no sign change, neither.
21. Second derivative testIf f'(c) = 0 and f''(c) > 0, local min (concave up); if f''(c) < 0, local max. If f''(c) = 0 the test is inconclusive.
22. What is an inflection point?A point on the graph where concavity changes (f'' changes sign). f''(c) = 0 alone is not enough.
23. Closed interval method for absolute extremaFor continuous f on [a, b], evaluate f at the critical numbers inside and at the endpoints a and b; the largest is the absolute max and the smallest the absolute min.
24. State the Mean Value Theorem.If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f'(c) = [f(b) - f(a)] / (b - a).
25. Maximize the area of a rectangle with perimeter 40.With 2x + 2y = 40, y = 20 - x and A = x(20 - x). A' = 20 - 2x = 0 gives x = 10, so a 10 by 10 square with area 100.
26. L'Hopital's rule: when can you use it?For limits of indeterminate form 0/0 or infinity/infinity, lim f/g = lim f'/g' (if the latter exists). Differentiate top and bottom separately, not with the quotient rule.
27. Linear approximation formulaL(x) = f(a) + f'(a)(x - a), the tangent line used to approximate f near a. Example: sqrt(4.1) is about 2 + (1/4)(0.1) = 2.025.
28. Position, velocity, acceleration relationshipv(t) = s'(t) and a(t) = v'(t) = s''(t). An object speeds up when v and a have the same sign.
29. Common pitfall: is d/dx [e^(2x)] = e^(2x)?No. By the chain rule it is 2e^(2x). Forgetting the inner derivative is a frequent error.
30. When is a function not differentiable at a point?At corners or cusps (like |x| at 0), vertical tangents, and discontinuities. Differentiable implies continuous, but not the reverse.

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