University:
University of North Georgia
Course:
MATH 1450 | Calculus I
Academic year:

2019
Views:

172

Pages:

2
Author:

5CelestialChronicle

AP Calc BC Review - Quiz Ch 4 1. Find the maximum and minimum values of f(x) = x^(2/3)(20-x) on [-1, 20] and state where these values occur. f'(x) = (2/3)x^(-1/3)(20-x) + x^(2/3)(-1) = (40-2x)/(3x^(1/3)) - x^(2/3) = (-5x+40)/(3x^(1/3)) Critical Points: x = 8, x = 0 Endpoints: x = -1, x = 20 f(0) = 0^(2/3)(20-0) = 0 f(8) = 8^(2/3)(20-8) = 48 f(-1) = (-1)^(2/3)(20+1) = 21 f(20) = 20^(2/3)(20-20) = 0 Max of 48 at x = 8 Min of 0 at x = 0, 20 2. Find the lengths of the sides of the isosceles triangle with perimeter 12 and maximum area. A = (1/2)xh h = √(36 - 6x + (x^2)/4) A' = (1/2)x * (1/2)(36-6x)^(-1/2) * (-6) + √(36-6x) * (1/2) = (-6x + 2√(36-6x))/(4√(36-6x)) = (-9x + 36)/(2√(36-6x)) Max when x = 4 and y = 4 3. A box-shaped wire frame consists of two identical wire squares whose vertices are connected by four straight wires of equal length. If the frame is to be made from a wire of length L, what should the dimensions be (in terms of L) to obtain a box of greatest volume? V = x²y L = 8x + 4y y = (L-8x)/4 V = x²(L-8x)/4 V' = (Lx - 8x²)/4 2Lx - 24x² = 0 2x(L - 12x) = 0 x = 0, x = L/12 Max V when x = L/12 and y = L/12 4. Let f(x) = x⁴ + ax². Find the value of a if: a) f has a local minimum at x = 5 f'(x) = 4x³ + 2ax f'(5) = 4(5)³ + 2a(5) = 0 500 + 10a = 0 a = -50 b) f has a point of inflection at x = 2 f''(x) = 12x² + 2a f''(2) = 12(2)² + 2a = 0 48 + 2a = 0 a = -24 Let f be a function defined on the closed interval -5 ≤ x ≤ 5 with f(1) = 3. The graph of f', the derivative of f, consists of two semicircles and two line segments, as shown above. a) For 1 -5 < x < 5, find all values of x at which f has a relative maximum. Justify your answer. max at x = -3 and x = 4. f' changes from + to - at x = -3 and x = 4. b) For -5 < x < 5, find all values of x at which the graph of f has a point of inflection. Justify your answer. Points of inflection at x = -4, -1, 2. f'' changes from + to - or vice versa. (or f' changes from increasing to decreasing or vice versa) c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning. -5 < x < -4 and 1 < x < 2. f'' is + and f' is also +

Review for Quiz Chapter 4 Worksheet Key

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