University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

59

Pages:

4
Author:

Luciano Kirby

Question 1 What's the derivative of tanh(x)? Answer We can write like that: tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)) So, we have: d/dx tanh(x) = [(e^x + e^(-x))(e^x + e^(-x)) - (e^x - e^(-x))(e^x - e^(-x))] / (e^x + e^(-x))² = [(e^x + e^(-x))² - (e^x - e^(-x))²] / (e^x + e^(-x))² = 1 - (e^x - e^(-x))² / (e^x + e^(-x))² = 1 - tanh²(x) Question 2 Evaluate arcsin(1/2) Answer sin(x) = 1/2 x = π/6 or 30 degree. Trig unit circle gives another arc x that has the same sin value x = (180° - 30°) = 150° So, the answer is 30° and 150°. Question 3 What does sec x equal? Answer sec θ = 1 / cos θ Question 4 Interval [0, 2π) 2cos²x - cos x = 0? Answer use the double angle formula: 2cos²x = 1 + cos(2x) Question 5 How do you verify the identity sin(2x) / sin(x) = 2 / sec(x) ? Answer 1 Have a look: sin(2x) = 2sin(x)cos(x) sec(x) = 1 / cos(x) 2sin(x)cos(x) / sin(x) = 2 / (1 / cos(x)) 2cos(x) = 2cos(x) Answer 2 Verify the following identity: sin(2x)/sin(x) = 2/sec(x) Eliminate the denominators on both sides. Cross multiply: sec(x) sin(2x) =? 2 sin(x) Express the left hand side in terms of sine and cosine. Write secant as 1/cos: (1/cos(x)) sin(2x) =? 2 sin(x) Eliminate the denominator on the left hand side. Multiply both sides by cos(x): sin(2x) = 2 cos(x) sin(x) Use the double angle identity on sin(2x) sin(2x) = 2sin(x) cos(x): 2 cos(x) sin(x) =? 2 cos(x) sin(x) Come to a conclusion. The left hand side and right hand side are identical: (identity has been verified)