MAT135 Exam Review: Limits, Derivatives, Optimization and Differential Equations

Terms in this set (32)

1. When does lim (x→a) f(x) exist?When the left-hand limit and the right-hand limit both exist and are equal. The value of f(a) itself does not matter; f need not even be defined at a.
2. Evaluate lim (x→2) (x^2 - 4)/(x - 2).Factor: (x - 2)(x + 2)/(x - 2) = x + 2 for x ≠ 2, so the limit is 4.
3. What is lim (x→0) sin(x)/x?1, with x in radians. This standard limit is used to derive that d/dx [sin x] = cos x.
4. What three conditions make f continuous at x = a?f(a) is defined, lim (x→a) f(x) exists, and lim (x→a) f(x) = 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 f(c) = N for some c in (a, b). It is often used to show an equation has a root, for example when f(a) < 0 < f(b).
6. Find the horizontal asymptote of f(x) = (3x^2 + 1)/(2x^2 - 5).The degrees of the top and bottom are equal, so the asymptote is the ratio of the leading coefficients: y = 3/2.
7. State the limit definition of the derivative.f'(x) = lim (h→0) [f(x + h) - f(x)]/h. It is the slope of the tangent line, or the instantaneous rate of change of f at x.
8. Does differentiable imply continuous? Does continuous imply differentiable?Differentiable implies continuous. The converse is false: f(x) = |x| is continuous at 0 but not differentiable there, because it has a corner.
9. Basic derivative rules: x^n, e^x, ln x, a^xd/dx [x^n] = n·x^(n-1). d/dx [e^x] = e^x. d/dx [ln x] = 1/x. d/dx [a^x] = a^x·ln a.
10. State the product rule and the quotient rule.(fg)' = f'g + fg'. (f/g)' = (f'g - fg')/g^2. A common mistake is writing (fg)' = f'·g', which is wrong.
11. State the chain rule and find d/dx [sin(x^3)].d/dx [f(g(x))] = f'(g(x))·g'(x). So d/dx [sin(x^3)] = cos(x^3)·3x^2.
12. 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.
13. 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.
14. Differentiate y = x^x.Take logs: ln y = x·ln x. Differentiate: y'/y = ln x + 1. So y' = x^x·(ln x + 1).
15. A circle's radius grows at 2 cm/s. How fast is its area growing when r = 5 cm?A = πr^2, so dA/dt = 2πr·(dr/dt) = 2π(5)(2) = 20π cm^2/s, about 62.8 cm^2/s.
16. Linear approximation formula; use it to estimate sqrt(4.1).L(x) = f(a) + f'(a)(x - a). With f(x) = sqrt(x) and a = 4: f'(4) = 1/(2·2) = 1/4, so sqrt(4.1) ≈ 2 + (1/4)(0.1) = 2.025.
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 maxima and minima can occur only at critical points (or at the endpoints of an interval).
18. First derivative testIf f' changes from + to - at c, f has a local maximum at c. If f' changes from - to +, it has a local minimum. If f' does not change sign, c is not a local extremum.
19. What does f'' tell you, and what is an inflection point?If f'' > 0, f is concave up; if f'' < 0, f is concave down. An inflection point is where the concavity changes. For example, f(x) = x^3 has one at x = 0.
20. How do you find the absolute max and min of a continuous function on [a, b]?By the Extreme Value Theorem both exist. 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.
21. State the Mean Value Theorem.If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) - f(a)]/(b - a) for some c in (a, b). So at some instant, the instantaneous rate equals the average rate.
22. A rectangle has perimeter 40. What dimensions maximize its area?With width x and length 20 - x, A(x) = x(20 - x). Then A'(x) = 20 - 2x = 0 gives x = 10. A 10 x 10 square gives the maximum area of 100.
23. State L'Hôpital's rule.If lim f(x)/g(x) has the form 0/0 or ∞/∞, it equals lim f'(x)/g'(x) when that limit exists. For example, lim (x→0) (e^x - 1)/x = lim e^x/1 = 1.
24. How do you check whether a function solves a differential equation?Substitute the function and its derivatives into the equation and confirm both sides are equal for all t. For example, y = e^(3t) solves y' = 3y because y' = 3e^(3t) = 3y.
25. Solve dy/dt = ky with y(0) = y0.y(t) = y0·e^(kt). If k > 0 the solution grows exponentially; if k < 0 it decays exponentially.
26. How do you find doubling time and half-life for exponential models?For growth y = y0·e^(kt), the doubling time is ln(2)/k. For decay y = y0·e^(-kt), the half-life is ln(2)/k.
27. Find and classify the equilibria of dy/dt = y(3 - y).Set dy/dt = 0: y = 0 and y = 3. Near y = 0 the solution moves away (unstable). Near y = 3, dy/dt > 0 just below and < 0 just above, so solutions move toward 3 (stable).
28. What is a slope field?A grid of short line segments whose slope at each point (t, y) equals dy/dt given by the differential equation. Solution curves follow the segments, which shows their behaviour without solving the equation.
29. Euler's method: give the formula and apply two steps to dy/dt = y, y(0) = 1, h = 0.5.y_(n+1) = y_n + h·f(t_n, y_n). Step 1: 1 + 0.5(1) = 1.5. Step 2: 1.5 + 0.5(1.5) = 2.25. So y(1) ≈ 2.25, while the exact value is e ≈ 2.718. Euler's method underestimates here because the solution is concave up.
30. Logistic equation: what do r and K mean, and when is growth fastest?dP/dt = rP(1 - P/K). r is the intrinsic growth rate and K is the carrying capacity, a stable equilibrium. Growth is fastest at P = K/2.
31. Newton's law of coolingdT/dt = -k(T - Ta), where k > 0 and Ta is the ambient temperature. The solution is T(t) = Ta + (T0 - Ta)e^(-kt), so the temperature approaches Ta exponentially.
32. Find d/dx [e^(2x)] and d/dx [ln(5x)].d/dx [e^(2x)] = 2e^(2x), by the chain rule. d/dx [ln(5x)] = 5/(5x) = 1/x.