Calculus I (MAT 145) Dr. Day Friday Feb 5, 2016 The Tangent-Line Slope Function: Also Known as The Derivative (2.7 and 2.8) Friday, February 5, 2016 MAT 145

Here is the graph of a function f. Use it to sketch the graph of f ’. Friday, February 5, 2016 MAT 145

Here is a graph of the function y = g(x) Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination. Friday, February 5, 2016 MAT 145

Here is the graph of the function y = |x|. Why does the derivative NOT exist at x = 0? Friday, February 5, 2016 MAT 145

Three situations for which a derivative DOES NOT EXIST! Friday, February 5, 2016 MAT 145

For each graphed function, state points at which the function is NOT differentiable. Explain your choices! Friday, February 5, 2016 MAT 145

Match each function, a-d, with its derivative, I-IV. Friday, February 5, 2016 MAT 145

Identify each curve. Explain your choices. Here are the graphs of four functions. One repre- sents the position of a car as it travels, another represents the velocity of that car, a third repre- sents the acceleration of the car, and a fourth graph represents the jerk for that car. Identify each curve. Explain your choices. Friday, February 5, 2016 MAT 145

Derivative Rules Function type Derivative Rule Constant for constant c Power for any real number n Product of constant and functions for constant c & function f(x) Sum or difference of functions for functions f(x) and g(x) Natural exponential function Exponential Functions Monday, February 8, 2016 MAT 145

Using Derivative Patterns For f(x) = 2x2 – 3x + 1: Calculate f’(x). Determine an equation for the line tangent to the graph of f when x = −1. Determine all values of x that lead to a horizontal tangent line. Determine all ordered pairs of f for which f’(x) = 1. Monday, February 8, 2016 MAT 145

Using Derivative Patterns MAT 145 Suppose s(x), shown below, represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds, for x > 0. Calculate the object’s velocity and acceleration functions. Is the object moving left or right at time x = 1? Justify. Determine the object’s velocity and acceleration at time x = 2. Based on those results, describe everything you can about the object’s movement at that instant. Write an equation for the tangent line to the graph of s at time x = 1. Monday, February 8, 2016

Using Derivative Patterns Determine the equation for the line tangent to the graph of g at x = 4. Determine the equation for the line normal to the graph of g at x = 1. At what points on the graph of g, if any, will a tangent line to the curve be parallel to the line 3x – y = –5? Monday, February 8, 2016 MAT 145