University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

461

Pages:

3
Author:

nesikhiyawl0

Question 1 find the exact value of tan(π/4 - π/3) Answer Trigonometry: tan(a-b) We know tan(a - b) = (tan a - tan b) / (1 + tan a tan b) tan(π/4 - π/3) = tan(240°) = tan(90-2+60°) = -tan(60°) = -√3 tan(π/4) = 1 tan(π/4 - π/3) = (tan(π/4) - tan(π/3)) / (1 + tan(π/4)tan(π/3)) = (1 - √3) / (1 + 1 * √3) = (1- √3) / (1 + √3) = (√3 - 1)(√3 - 1) / (√3 + 1)(√3 - 1) = (3 - 2√3 + 1) / (3 - 1) = (4 - 2√3) / 2 = 2 - √3 = 0.2679 Question 2 Evaluate the trigonmetric function at the given real number: csc(5π/4) Answer 3π/2 < 5π/4 < π Thus, the angle lies in third quadrant. Therefore, csc(5π/4) = 1 / sin(x) (a) sin(x) = sin(π + π/4) sin(π + π/4) = -sin(π/4) -sin(π/4) = -1/√2 csc(5π/4) = 1 / (-1/√2) Substitute the value in (a) csc(5π/4) = -√2 csc(5π/4) = -√2 csc(5π/4) = (-√2 * √2) / √2 csc(5π/4) = -2/√2 = -√2 Question 3 Express sin (arccos (2/x)) in terms of x. Answer sin (arccos(2/x)) sin (arccos(2/x)) = sin (arcsin(√(1 - (2/x)²))) sin (arccos(2/x)) = sin (arcsin(√(1 - 4/x²))) sin (arccos(2/x)) = sin (arcsin(√(x²-4)/x²)) Since, sin (arcsin(√(x²-4)/x²)) = √(x²-4)/x² Then, sin (arccos(2/x)) = √(x²-4)/x Question 4 y = x²e^x - 2xe^x + 4e^x The derivative of the function is? Answer y = x²e^x - 2xe^x + 4e^x d/dx (e^ax) = ae^ax y' = d/dx(x²e^x - 2xe^x + 4e^x) dy/dx = (x²e^x - 2xe^x + 4e^x)' dy/dx = (x² - 2x)'(e^x) + (e^x)'(x² - 2x) + d/dx(4e^x) dy/dx = (2x - 2)e^x + (e^x)(x² - 2x) + 4e^x dy/dx = (x² - 2x)e^x + (2x - 2)e^x + 4e^x dy/dx = (x² - 2x)e^x + (2x - 2)e^x + 4e^x dy/dx = (x² - 2x + 2x - 2 + 4)e^x dy/dx = (x² + 2)e^x Question 5 Convert the following angle from degrees to radians: 240° Answer To convert from degrees to radians, multiply by π/180 240° * π/180 = 240π/180 = (60 * 4π)/(60 * 3) = 4π/3