Monday, September 7, 2015MAT 145. Monday, September 7, 2015MAT 145.

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Presentation transcript:

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 The concept of... LIMIT!

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 Sketch the graph of by finding its intercepts and its limits as and as.

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 Here is a graph of a function f. At which values is f discontinuous? Why?

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 Removable Discontinuity Infinite Discontinuity Jump Discontinuity Removable Discontinuity

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!

Monday, September 7, 2015MAT 145 Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145 A B C D E

Monday, September 7, 2015MAT 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, September 7, 2015MAT 145 Calculate the slope of f(x) = x 2 at x = a.

Monday, September 7, 2015MAT 145 We call this slope calculation the derivative of f at x = a.

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 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 7, 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?

Monday, September 7, 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?

Monday, September 7, 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?

Monday, September 7, 2015MAT 145 Can we create a derivative function f that will be true for any x value where a derivative exists?

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 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 7, 2015MAT 145 Calculate the derivative function, f ’(x), for f(x) = x 2. Use the limit definition of the derivative.

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145

Monday, September 7, 2015MAT 145