University:
Santa Ana College
Course:
MATH 180 | Single Variable Calculus I
Academic year:

2023
Views:

227

Pages:

5
Author:

gylchwylenal

Exam Math 180 Name: 1) Find each limit of f(x) or state it does not exist: a) lim f(x) = -2 b) lim f(x) = 2 c) lim f(x) = DNE d) lim f(x) = -1 e) lim f(x) = ∞ 2) Find the limit algebraically, if it exists. (Give exact values): a) lim (x^4 - 2x^2 + 3) = 227 b) lim (x+4)/(x^2 - x - 20) = -1/9 c) lim (3x^2 - 7x - 6)/(x - 3) = 11 3) Graph the function f(x) then use the graph to answer the questions: [f(x)=\begin{cases}\frac{1}{2}x^{3}-5&for~x\le-2\ |x|+2&for~x>-2.\end{cases}] a) List the points on the graph at which the function is not differentiable: at x=-2 and x=0 (sharp point) b) Is f(x) continuous? NO If not, where is it discontinuous? x=-2 4) Find the derivative of each. Answers may not contain fractions or negative exponents: a) y=6x^5+5x^3-3x+8 y'=30x^4+15x^2-3 b) y=x^{2/3} y'=2/3x^(-1/3) c) y=5/x^2-6x y'=-10x^(-3)-6 d) y=10√x+3√x y'=5x^(-1/2)+3x^(-1/2) e) y=2/3x^9-1/4x^8+3x^7-100 y'=6x^8-2x^7+21x^6 5) Find the derivative of each. Answers may not contain fractions or negative exponents: a) y = (5-3x)^4 (chain rule) y' = 4(5-3x)^3(-3) y' = -12(5-3x)^3 b) y = (3x-5)/(2x+7) (quotient rule) y' = [3(2x+7) - 2(3x-5)]/(2x+7)^2 y' = 31/(2x+7)^2 c) y = √(4x^2-6x) (chain rule) y' = 1/2(4x^2-6x)^(-1/2)(8x-6) y' = (4x-3)/(2√(4x^2-6x)) d) y = x^5(3x+1)^6 (product rule) y' = 5x^4(3x+1)^6 + 6(3x+1)^5(3)x^5 y' = x^4(3x+1)^5[5(3x+1) + 18x] y' = x^4(3x+1)^5(33x+5) 6) Algebraically find the equation of the tangent line to f(x) = x + 4/x at x = 4: Point: (4, 5/2) f'(x) = 1 - 4x^(-2) f'(4) = 1 - 1/4 = 3/4 Equation of tangent line: y - 5/2 = 3/4(x - 4) y = 3/4x + 2 7) Tam's T-shirt Co. Cost function: C(x) = 0.04x^2 + 13x + 1200 Revenue function: R(x) = 43x - 0.02x^2 a) Find the profit function: P(x) = R(x) - C(x) P(x) = -0.06x^2 + 30x - 1200 b) Evaluate and Interpret P(50): P(50) = -0.06(50)^2 + 30(50) - 1200 P(50) = 150 When they produce and sell 50 t-shirts, the profit is $150. c) Evaluate and Interpret P'(100): P'(x) = -12x + 30 P'(100) = -12(100) + 30 P'(100) = 18 When they sell 100 t-shirts, the profit is increasing at a rate of $18 per shirt. 8) The total number of dropouts for grades 10-12 during the 1970s and 1980s: D(x) = -2.1x^2 + 26.5x + 602 a) Determine and Interpret D(5): D(5) = -2.1(5)^2 + 26.5(5) + 602 D(5) = 682 In 1975, the total number of dropouts was 682,000 students. b) Find D'(x): D'(x) = -4.2x + 26.5 c) Determine and Interpret D'(2) and D'(10): D'(2) = 18.1 In 1972, the number of dropouts was increasing at a rate of 18,100 students per year. D'(10) = -15.5 In 1980, the number of dropouts was decreasing at a rate of 15,500 students per year.

Math 180 Exam Key 7 Solutions

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