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I’m going nuts over derivatives!!! 2.1 The Derivative and the Tangent Line Problem

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Calculus grew out of 4 major problems that European mathematicians were working on in the seventeenth century. 1.The tangent line problem 2.The velocity and acceleration problem 3.The minimum and maximum problem 4.The area problem

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The tangent line problem (c, f(c)) secant line f(c+ ) – f(c) x (c, f(c)) is the point of tangency and is a second point on the graph of f.

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The slope between these two points is Definition of Tangent Line with Slope m

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Find the slope of the graph of f(x) = x 2 +1 at the point (-1,2). Then, find the equation of the tangent line. (-1,2)

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Therefore, the slope at any point (x, f(x)) is given by m = 2x What is the slope at the point (-1,2)? m = -2 The equation of the tangent line is y – 2 = -2(x + 1) f(x) = x 2 + 1

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The limit used to define the slope of a tangent line is also used to define one of the two funda- mental operations of calculus --- differentiation Definition of the Derivative of a Function f’(x) is read “f prime of x” Other notations besides f’(x) include:

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Find f’(x) for f(x) = and use the result to find the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)? 1

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Therefore, at the point (1,1), the slope is ½, and at the point (4,2), the slope is ¼. What happens at the point (0,0)? The slope is undefined, since it produces division by zero. 1 2 3 4

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Find the derivative with respect to t for the function

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Theorem 3.1 Alternate Form of the Derivative The derivative of f at x = c is given by (c, f(c)) cx (x, f(x))

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Derivative from the left and from the right. Example of a point that is not differentiable. is continuous at x = 2 but let’s look at it’s one sided limits. 1

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The 1-sided limits are not equal., x is not differentiable at x = 2. Also, the graph of f does not have a tangent line at the point (2, 0). A function is not differentiable at a point at which its graph has a sharp turn or a vertical tangent line(y = x 1/3 or y = absolute value of x). Differentiability can also be destroyed by a discontinuity ( y = the greatest integer of x).

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THE DERIVATIVE AND THE TANGENT LINE PROBLEM

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

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