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

2024
Views:

401

Pages:

3
Author:

nesikhiyawl0

Question 1 Find the exact value of the trigonometric function cos(9π / 4) Answer Rewriting the given trigonometric function, cos(9π / 4) = cos(8π / 4 + π / 4) = cos(2π + π / 4) We know that (2π + θ) always lies in first quadrant and in first quadrant all trigonometric functions are positive. So applying above rule for the given trigonometric function, we get cos(2π + π / 4) = cos(π / 4) = 1 / √2 Hence, exact value of the given trigonometric function is 1 / √2 Question 2 Find the exact value of the trigonometric function sec(-9π / 4). Answer We know that sec(-x) = sec(x) Therefore, sec(-9π / 4) = sec(9π / 4) Rewriting the above trigonometric function, sec(9π / 4) = sec(8π / 4 + π / 4) = sec(2π + π / 4) We know that 2π + θ lies in first quadrant and in first quadrant all trigonometric functions are positive. Therefore, sec(2π + π / 4) = sec(π / 4) = √2 Hence, exact value of given trigonometric function is √2 Question 3 Find the derivative of y = sin(πx) Answer If you're in a different section of your course, please let me know in a comment. This is an implementation of the chain rule. Keep in mind that the chain rule essentially states that you must first multiply the derivatives of the "inside" and "outside" functions independently. The outside function here is f(x) = sin(x). Keep in mind that its derivative cos(x) The inside function is g(x) = πx. The derivative of this is clearly π. Putting these together we get f' = cos(πx) * π = πcos(πx) where the πx in the cos function is derived from the requirement that, when multiplying, the identical argument from the external function be inserted. If that doesn't make sense, please let me know. Question 4 Find the differential of the following functions: y = tan(√3t) y = 5 - v² / 5 + v² Answer y = tan(√3t) y'(t) = sec²(√3t) * d/dt(√3t) = sec²(√3t) * (√3 / 2√t) = √3sec²(√3t) / 2√t y(t) = d/dv(5 - v²/5 + v²) = 0 - (2v(5 + v²) - v²(2v)) / (5 + v²)² = - (10v + 2v³ - 2v³) / (5 + v²)² = -10v / (5 + v²)² Question 5 s(x) = 3600 / (60 + x) = 3600(60 + x)⁻¹ is a function that gives average speed. It takes 60+x seconds to travel one km. Find average speed if one km is traveled in 53 seconds. Answer s(x) = 3600 / (60 + x) d/dx (s(x)) = d/dx (3600 / (60 + x)) = 3600 * d/dx(1/(60 + x)) = -3600 / (60 + x)² Approximate speed equation: L(x) ≈ s(a) + s'(x - a) Where a = 0 and x = 53. L(x) ≈ s(0) + s'(53 - 0) = 3600 / (60+0) - 3600 / (60 + 53)² * 53 = 3600/60 - 3600 / 113² * 53 = 60 - 3600 * 53 / 12769 = 60 - 0.2819 = 59.718 The approximate average speed is 59.718 km/hr.