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Section 3.1 The Derivative and the Tangent Line Problem

Section 3.1 The Derivative and the Tangent Line Problem. Q. P. f ( x 1 ). f ( x 2 ). x 1. x 2. Q. Q. f ( x 2 ). Q. f ( x 2 ). f ( x 2 ). x 2. x 2. x 2. Secant and Tangent Lines. Secant Line – A line passing through two points on a graph of a function.

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Section 3.1 The Derivative and the Tangent Line Problem

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  1. Section 3.1 The Derivative and the Tangent Line Problem

  2. Q P f(x1) f(x2) x1 x2 Q Q f(x2) Q f(x2) f(x2) x2 x2 x2 Secant and Tangent Lines Secant Line – A line passing through two points on a graph of a function. Tangent Line – A line that touches the graph at a point. The tangent line may cross the graph at other points depending on the graph. P f(x1) x1 http://www.slu.edu/classes/maymk/Applets/SecantTangent.html

  3. Q P f(a) f(x) a x Slope of the Secant Line Now let x1 = a and x2 = x. This changes the formula to:

  4. Slope of the Tangent Line To determine the slope of the tangent line, let xapproach a. In terms of limits, we are finding the slope of the secant line as x→a.

  5. Q P f(x1) f(x2) x1 x2 Q P f(a) f(a+h) a a + h Slope of the Tangent Line (another form) Now let x1 = a and x2 = a+h, where h is the distance from x1 to x2. This changes the formula to:

  6. Secant line Q Tangent f (a + h) – f (a) P Let h approach 0 h

  7. Q P f(a) f(a+h) a a + h Slope of the Tangent Line ← meansh → 0

  8. Example • Find an equation of a tangent line to f (x) = x2at • x = 3 • x = -1 • x = 4

  9. Drawing tangent lineunder the “draw” menu. Finding slope of the tangentunder the “Calc” menu. slope tangent equation

  10. The Derivative For y = f (x), we define thederivative of f at x, denotedf ’ (x), to be if the limit exists. If f’(x) exists at a point a, we say f is differentiableat a.

  11. Interpretations of the Derivative • If f is a function, then f ’ (x) is a new function that represents • the slope of the tangent line to the graph of f at x. • the instantaneous rate of change of f (x) with respect to x. • the velocity of the object at time x if f (x) is the position function of a moving object.

  12. Example Find the derivative of f (x) = x 2 – 3x.

  13. Examples Notations is called Leibniz Notation Find of f (x) = 5/x Find f’(2) where f (x) = | x – 2|

  14. Nonexistence of the Derivative • Some of the reasons why the derivative of a function may not exist at x = a are • The graph of f is not continuous at x = a • The graph of f has a sharp corner at x = a • The graph of f has a vertical tangent at x = a. • If f is differentiable at a, the graph of f is “smooth” at a.

  15. Theorem If f is differentiable at a, then f is continuous at a. Caution:If f is continuous at x = a, then it is not necessarily differentiable at x = a.

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