Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145 Find an equation of the tangent line to the parabola y = x 2 at the point P(1, 1).

Monday, February 1, 2016MAT 145 A B C D E

Monday, February 1, 2016MAT 145 Calculate the slope at x = −2. Calculate the slope at x = −1. Calculate the slope at x = 0. Calculate the slope at x = a.

Monday, February 1, 2016MAT 145 Calculate the slope of f(x) = x 2 at x = a.

Monday, February 1, 2016MAT 145 We call this slope calculation the derivative of f at x = a.

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 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, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145

Monday, February 1, 2016MAT 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.

Friday, Sept 25, 2015MAT 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?

Friday, Sept 25, 2015MAT 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.

Friday, Sept 25, 2015MAT 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?

Friday, Sept 25, 2015MAT 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.

Friday, Sept 25, 2015MAT 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, February 1, 2016MAT 145

Monday, February 1, 2016MAT 145 Find an equation of the tangent line to the parabola y = x 2 at the point P(1, 1).

Monday, February 1, 2016MAT 145 Can we create a derivative function f that will be true for any x value where a derivative exists?

Monday, February 1, 2016MAT 145 Calculate the derivative function, f ’(x), for f(x) = x 2. Use the limit definition of the derivative.

Monday, February 1, 2016MAT 145

Wednesday, February 3, 2016MAT 145 Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination.

Wednesday, February 3, 2016MAT 145 Here is the graph of the function y = |x|. Why does the derivative NOT exist at x = 0?

Wednesday, February 3, 2016MAT 145 Three situations for which a derivative DOES NOT EXIST!

Wednesday, February 3, 2016MAT 145 Three situations for which a derivative DOES NOT EXIST!

Wednesday, February 3, 2016MAT 145 For each graphed function, state points at which the function is NOT differentiable. Explain your choices!

Wednesday, February 3, 2016MAT 145 Match each function, a-d, with its derivative, I-IV.

Wednesday, February 3, 2016MAT 145 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.

Wednesday, February 3, 2016 MAT 145 Here is the graph of a function f. Use it to sketch the graph of f ’.