§ 2.2 The First and Second Derivative Rules

Section Outline First Derivative Rule Second Derivative Rule

First Derivative Rule

First Derivative Rule EXAMPLE Sketch the graph of a function that has the properties described. f (-1) = 0; for x < -1; for x > -1. SOLUTION The only specific point that the graph must pass through is (-1, 0). Further, we know that to the left of this point, the graph must be decreasing ( for x < -1) and to the right of this point, the graph must be increasing ( for x > -1). Lastly, the graph must have zero slope at that given point ( ).

Second Derivative Rule

First & Second Derivative Scenarios

First & Second Derivative Rules EXAMPLE Sketch the graph of a function that has the properties described. f (x) defined only for x ≥ 0; (0, 0) and (5, 6) are on the graph; for x ≥ 0; for x < 5, , for x > 5. SOLUTION The only specific points that the graph must pass through are (0, 0) and (5, 6). Further, we know that to the left of (5, 6), the graph must be concave down ( for x < 5) and to the right of this point, the graph must be concave up ( for x > 5). Also, the graph will only be defined in the first and fourth quadrants (x ≥ 0). Lastly, the graph must have positive slope everywhere that it is defined.

First & Second Derivative Rules CONTINUED

First & Second Derivative Rules EXAMPLE Looking at the graphs of and for x close to 10, explain why the graph of f (x) has a relative minimum at x = 10.

First & Second Derivative Rules CONTINUED SOLUTION At x = 10 the first derivative has a value of 0. Therefore, the slope of f (x) at x = 10 is 0. This suggests that either a relative minimum or relative maximum exists on the function f (x) at x = 10. To determine which it is, we will look at the second derivative. At x = 10, the second derivative is above the x-axis, suggesting that the second derivative is positive when x = 10. Therefore, f (x) is concave up when x = 10. Since at x = 10, f (x) has slope 0 and is concave up, this means that the f (x) has a relative minimum at x = 10.

First & Second Derivative Rules EXAMPLE After a drug is taken orally, the amount of the drug in the bloodstream after t hours is f (t) units. The figure below shows partial graphs of the first and second derivatives of the function.

First & Second Derivative Rules CONTINUED (a) Is the amount of the drug in the bloodstream increasing or decreasing at t = 5? (b) Is the graph of f (t) concave up or concave down at t = 5? (c) When is the level of the drug in the bloodstream decreasing the fastest? SOLUTION (a) To determine whether the amount of the drug in the bloodstream is increasing or decreasing at t = 5, we will need to consider the graph of the first derivative since the first derivative of a function tells how the function is increasing or decreasing. At t = 5 the value of the first derivative is -4. Therefore, the value of the first derivative is negative at t = 5. Therefore, the function is decreasing at t = 5.

First & Second Derivative Rules CONTINUED (b) To determine whether the graph of f (t) is concave up or concave down at t = 5, we will need to consider the graph of the second derivative at t = 5. At t = 5, the value of the second derivative is 0.5. Therefore, the value of the second derivative is positive at t = 5. Therefore, the function is concave up at t = 5. (c) To determine when the level of the drug in the bloodstream is decreasing the fastest, we need to determine when the first derivative is the smallest. This occurs when t = 4.