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Tangent Lines and Derivatives

Learn the concept of tangent lines to curves at points, derivative as slope, differentiability, continuity, & vertical tangent lines in calculus. Practice & quiz included.

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Tangent Lines and Derivatives

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  1. Lesson 2.1 The Derivative and the Tangent Line Problem Quiz

  2. . . P What does it mean to say that a line is tangent to a curve at a point? For a circle, the tangent line at a point P is the line that is perpendicular to the radial line at point P. For a general curve, however, the problem is more difficult.

  3. . (c+x, f(c+ x) . y --------------- (c, f(c)) ------------- x Essentially, the problem of finding the tangent line at a point P boils down to the problem of finding the slope of the tangent line at point P. You can approximate this slope using a secant line through two points on the curve.

  4. 2.1 The Derivative and the Tangent Line Problem c=2

  5. 2.1 The Derivative and the Tangent Line Problem The slope of a function is its derivative.

  6. 2.1 The Derivative and the Tangent Line Problem

  7. 2.1 The Derivative and the Tangent Line Problem

  8. 2.1 The Derivative and the Tangent Line Problem

  9. 2.1 The Derivative and the Tangent Line Problem

  10. 2.1 The Derivative and the Tangent Line Problem

  11. 2.1 The Derivative and Tangent Line Problem AP EXAM

  12. Differentiability and continuity The following alternative limit form of the derivative is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is

  13. 2.1 The Derivative and the Tangent Line Problem Graph by Hand

  14. 2.1 The Derivative and the Tangent Line Problem Vertical Tangent Line If a function is continuous at a point c and , then x = c is a vertical tangent line for the function.

  15. 2.1 The Derivative and the Tangent Line Problem HW 2.1/3,4,5-15odd,16,21,24,25,27-32,33,35,37, 41,45,47,62

  16. Common Denominator

  17. Common Denominator Conjugate...

  18. Common Denominator

  19. 62.

  20. Quiz

  21. Yea! You finished the lesson! Now get to work!

  22. . If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at point (c, f(c)). (c+x, f(c+ x) y --------------- (c, f(c)) ------------- . x . (c+x, f(c+ x) . y ------------ (c, f(c)) ---------- . x (c, f(c)) lim f(c + x) – f(c) x = m x→0

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