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Warm –Up

Warm –Up. Write the equation of the line with a slope of -2 that passes through the point (5,-1). Find the instantaneous rate of change of the curve f(x)=3x 2 -2 at x=1. 3.4: Definition of the Derivative. Objectives: To find the slope of a tangent line to a curve

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Warm –Up

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  1. Warm –Up • Write the equation of the line with a slope of -2 that passes through the point (5,-1). • Find the instantaneous rate of change of the curve f(x)=3x2-2 at x=1.

  2. 3.4: Definition of the Derivative Objectives: To find the slope of a tangent line to a curve To write the equation of a tangent line To find the derivative

  3. TANGENT LINE • A line that touches a curve at a single point but no other points nearby • It indicates the direction of the curve P2 P1 The line through P1 is a tangent line. Line through P2 is not.

  4. TANGENT LINE AND SLOPE • Secant line: Goes through any 2 points on a curve • Slope of secant line gives average rate of change A (a, f(a)) B (a +h, f(a+h)) Slope between A and B: As B slides down the graph closer and closer to A, h→0. It gets closer to the tangent line at A!! B A

  5. Check it out….. • http://www.slu.edu/classes/maymk/Applets/SecantTangent.html

  6. Slope of Tangent Line (look familiar???!?!?!?) PROVIDED THE LIMIT EXISTS If the limit DNE, no tangent line at that point!

  7. The slope of a tangent line at a point … • Instantaneous rate of change of y with respect to x at that point • Slope of the curve at that point • Indicates direction of curve at that point • Captures the behavior of the function very close to the point (assuming the function approximates a line up close)

  8. Examples: A. Find the equation of the secant line of f(x)=x2 through points x=2 and x=4. B. Find the slope of the tangent line and the equation of the tangent line at x=2.

  9. Write the equation of the tangent line to the curve at x = 3. You need a slope and a point to write the equation of a line. What is the value of the function at x =3? Find the slope of the curve at that point: Write the equation:

  10. Write the equation of the tangent line to the curve f(x)=x2+3 at x = -2.

  11. At what point is the tangent line to f(x)=x2-2x-3 horizontal?

  12. This leads us to….. THE DERIVATIVE!!!!!!

  13. The derivative of a function f at x is Provided the limit exists.

  14. THE DERIVATIVE IS…. • A function, f’(x), with its own domain and range that may be different from f(x) • If f’(x) exists, the function is differentiable • Instantaneous rate of change of f with respect to x (ex. Marginal cost, revenue, etc) • Slope of the graph of f(x) at and point x=a • Slope of the tangent line at x = a • Awesome • Instantaneous velocity

  15. Notations: Ways to write the derivative of y=f(x)

  16. Let f(x)=x2+3x. Find f ‘(x).

  17. Let . Find f ‘(x), f ‘(-1), and f ‘(0), if it exists.

  18. Examples: 1. Let y=x3+4x. Find y’ and y’(2). 2. Let . Find and .

  19. ALTERNATIVE FORM: Provided the limit exists.

  20. Given f(x)=x2+2, Find f’(4).

  21. New notation for equation of tangent line at point (x1, f(x1)) Provided f’(x1) exists. Find the equation of the tangent line of at x=4.

  22. Derivatives do not exist at sharp corners, vertical tangent lines, or where the function itself is not defined. Examples: • y= | x | • f(x) = x1/3

  23. Existence of the Derivative f’(x) exists at a point when f(x): a.) is continuous b.) is smooth c.) Does not have a vertical tangent line f’(x) does NOT exist at a point if: a.) f is discontinuous b.) f has a sharp corner c.) f has a vertical tangent line

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