Essay

Math 137

Assignment 5

Due Friday, Oct 18th

1. Find the derivative of the following functions. ex (a) f (x) = 3x + 1 x1/2 + ex (b) g(x) = 1 − ex 2. Find the equation of the tangent line to the function at the given point. (a) f (x) = 3x2 + 1 at a = −2. (b) f (x) = xex at a = 0 x2 + 1

3. Use the definition of the derivative to prove the following. (a) If f (x) = x2 − x, then f (x) = 2x − 1. (b) If f (x) = (c) If f (x) = √

1 , x+1

then f (1) = − 1 . 4

√1 . 2 x+1

x + 1, then f (x) =

4. If the position function of an object is given by x(t) = 3t3 − t2 + 2t − 4, then find the velocity v(t) and acceleration a(t) functions of the object. x2 + 2x − 1 if x = 1 5. Let f (x) = . 2 if x = 1 Determine if f (1) exists. If it does exist, what is the value of f (1)? x2 + 3 if x ≤ 1 . 3 x −1 if x > 1 Determine if f (1) exists. If it does exist, what is the value of f (1)? √

6. Let f (x) =

7. Assume that m(t) is the mass of an object in kg at any time t in hours. (a) What are the units for the functions m (t) and m (t)? (b) What does it mean physically for the object if m (2) > 0? (c) What does it mean physically for the object if m (2) < 0? 8. Use the definition of the derivative to prove that if f is differentiable, then d d cf (x) = c f (x) dx dx