Calculus I (MAT 145) Dr. Day Monday September 18, 2017 Secants Slopes to Tangent Slope: Using Limits (2.7/2.8) The Definition of Derivative (2.7/2.8) Monday, September 18, 2017 MAT 145

C B A D E Monday, September 18, 2017 MAT 145

Another way to label points and calculate slope Let x = a + h and so the slope of the secant line PQ is Monday, September 18, 2017 MAT 145

We call this slope calculation the derivative of f at x = a. Monday, September 18, 2017 MAT 145

Monday, September 18, 2017 MAT 145

Monday, September 18, 2017 MAT 145

Monday, February 1, 2016 Monday, September 18, 2017 MAT 145 MAT 145

Monday, September 18, 2017 MAT 145

The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the slope function as the secant lines get closer and closer to being a tangent line. Monday, September 18, 2017 MAT 145

The value f ’(a) is called: the derivative of f at x = a, the instantaneous rate of change of f at x = a, the slope of f at x = a, and the slope of the tangent line to f at x = a. Monday, September 18, 2017 MAT 145

The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100? Monday, September 18, 2017 MAT 145

The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100? From the description of the context, the “rate units” are: feet per minute. The value 12 is an input variable, so we are looking at the precise instant that 12 minutes of travel has occurred, since some designated starting time when t = 0. S’ indicates rate of change of S, indicating we have information about how S is changing with respect to t, in feet per minute. The value 100 specifies the rate: 100 feet per minute. Putting it all together: At precisely 12 minutes into the trip, the object’s position is increasing at the rate of 100 feet per minute. Monday, September 18, 2017 MAT 145

The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5? Monday, September 18, 2017 MAT 145

The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5? From the description of the context, the “rate units” are: thousands of dollars per patient. The value 90 is an input variable, so we are looking at the precise instant when 90 patients are in the hospital. C’ indicates rate of change of C, indicating we have information about how C is changing with respect to p, in thousands of dollars per patient. The value 4.5 specifies the rate: 4.5 thousand dollars ($4500) per patient. Putting it all together: At precisely the instant that 90 patients are in the hospital, the cost per patient is increasing at the rate of $4500 per patient. Monday, September 18, 2017 MAT 145

The derivative in action! V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π? From the description of the context, the “rate units” are: cubic cm of volume per cm of radius. The value 3 is an input variable, so we are looking at the precise instant when the sphere’s radius is 3 cm long. V’ indicates rate of change of V, indicating we have information about how V is changing with respect to r, in cubic cm per cm. The value 36π specifies the rate: 36π cubic cm of volume per 1 cm of radius length. Putting it all together: At precisely the instant that the sphere has a radius length of 3 cm, the sphere’s volume is increasing at the rate of 36π cubic cm per cm of radius length. Monday, September 18, 2017 MAT 145

Tangent slope again Monday, September 18, 2017 MAT 145

Can we create a derivative function f that will be true for any x value where a derivative exists? Monday, September 18, 2017 MAT 145

Calculate the derivative function, f ’(x), for f(x) = x2 Calculate the derivative function, f ’(x), for f(x) = x2. Use the limit definition of the derivative. Monday, September 18, 2017 MAT 145

Monday, September 18, 2017 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. Monday, September 18, 2017 MAT 145

Here is the graph of the function y = |x|. Why does the derivative NOT exist at x = 0? Monday, September 18, 2017 MAT 145

Three situations for which a derivative DOES NOT EXIST! Monday, September 18, 2017 MAT 145

For each graphed function, state points at which the function is NOT differentiable. Explain your choices! Monday, September 18, 2017 MAT 145

Match each function, a-d, with its derivative, I-IV. Monday, September 18, 2017 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. Monday, September 18, 2017 MAT 145

Here is the graph of a function f. Use it to sketch the graph of f ’. Monday, September 18, 2017 MAT 145