Math 2250 Written HW #6 Solutions 1. Roughly speaking, we can approximate the human-equivalent age of a dog which is y years old by the function y2 √ h(y) = 16 y + . 15 This isn’t perfect, but it’s certainly more accurate than claiming that a 1-year-old dog is equivalent to a 7-year-old human. Anyway, assuming this function is correct: (a) What information does the function h0 (y) convey? To guide your thinking, consider the following: if h0 (2) ≈ 5.92, what does that really tell us about how a 2-year-old dog is aging? Answer: By definition, the derivative h0 (y) gives the rate of change of the function h(y). Recall that h(y) does the following: given a dog which is y years old, h(y) tells us the human-equivalent age of the dog. So the rate of change of that function tells us how fast the human-equivalent age of a dog is changing. For example, h0 (2) ≈ 5.92 tells us that when it is 2 years old a dog is aging by about 6 human years per actual year. In other words, a 2-year-old dog is aging 6 times faster than a human. Whereas, since 8 h0 (4) = 4 + 15 (as we’ll see below), a 4-year-old dog is aging about 4.5 times faster than a human. In general, h0 (y) tells us how many times faster a y-year-old dog is aging than a human. (b) What is h0 (4)? Answer: We can re-write h(y) as h(y) = 16y 1/2 + 1 2 y , 15 so by splitting up and using the Power Rule, we see that 1 −1/2 1 16 2y 8 2y 0 h (y) = 16 y + (2y) = √ + =√ + . 2 15 2 y 15 y 15 Therefore, 8 2·4 8 h0 (4) = √ + =4+ . 15 15 4 1 2. Consider the function f (x) = x2 e x . 1+x2 What is f 0 (x)? Answer: By the quotient rule, f 0 (x) = = = = = = d d (1 + x2 ) dx (x2 ex ) − x2 ex dx (1 + x2 ) (1 + x2 )2 d d (1 + x2 ) dx (x2 )ex + x2 dx (ex ) − x2 ex (2x) (1 + x2 )2 (1 + x2 )(2xex + x2 ex ) − x2 ex (1 + x2 )2 x 2 (2xe + x ex + 2x3 ex + x4 ex ) − 2x3 ex (1 + x2 )2 2xex + x2 ex + x4 ex (1 + x2 )2 xex (2 + x + x3 ) (1 + x2 )2 3. Find the derivative of the function g(x) = sin(x) cos(x). Answer: By the product rule, d d (sin(x)) cos(x) + sin(x) (cos(x)) dx dx = (cos(x)) cos(x) + sin(x)(− sin(x)) g 0 (x) = = cos2 (x) − sin2 (x). If we use the double-angle identity cos(2θ) = cos2 (θ) − sin2 (θ), we could also write g 0 (x) as g 0 (x) = cos(2x). 2