CHAPTER 3
The Derivative SECTION 3.1 3.1.1 Let f (x) = (a) (b) (c) (d)
1 . x2
Find the average rate of change of y with respect to x over the interval [2, 3]. Find the instantaneous rate of change of y with respect to x at the point x = 2. Find the instantaneous rate of change of y with respect to x at a general point x0 . Sketch the graph of y = f (x) together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
3.1.2 Let f (x) = x2 + 1. (a) (b) (c) (d)
Find the average rate of change of y with respect to x over the interval [−2, −1]. Find the instantaneous rate of change of y with respect to x at the point x = −2. Find the instantaneous rate of change of y with respect to x at a general point x0 . Sketch the graph of y = f (x) together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
3.1.3 Let f (x) = (a) (b) (c) (d)
Find the average rate of change of y with respect to x over the interval [3, 5]. Find the instantaneous rate of change of y with respect to x at the point x = 3. Find the instantaneous rate of change of y with respect to x at a general point x. Sketch the graph of y = f (x) together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
3.1.4 Let f (x) = (a) (b) (c) (d)
1 . x−2
1 . x+1
Find the average rate of change of y with respect to x over the given interval [1, 3]. Find the instantaneous rate of change of y with respect to x at the point x = 1. Find the instantaneous rate of change of y with respect to x at the general point x0 . Sketch the graph of y = f (x) together with the secant and tangent lines whose slopes are given by the results in parts (a) and (b).
3.1.5 Let f (x) =
2 . 3−x
(a) Find the slope of the tangent to the graph of f at a general point x0 using the method of Section 3.1. (b) Use the result in part (a) to find the slope of the tangent at x0 = 1. 3.1.6 Let f (x) =
3 . x−1
(a) Find the slope of the tangent to the graph of f at a general point x0 using the method of Section 3.1. (b) Use the result in part (a) to find the slope of the tangent at x0 = 4.
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