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2.1 The Derivative and the Tangent Line Problem (Part 1)

Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. 2.1 The Derivative and the Tangent Line Problem (Part 1). Great Sand Dunes National Monument, Colorado. Objectives. Find the slope of the tangent line to a curve at a point.

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2.1 The Derivative and the Tangent Line Problem (Part 1)

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  1. Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington 2.1 The Derivative and the Tangent Line Problem (Part 1) Great Sand Dunes National Monument, Colorado

  2. Objectives • Find the slope of the tangent line to a curve at a point. • Use the limit definition to find the derivative of a function.

  3. Origins of Calculus Calculus grew out of four major problems that European mathematicians were working on during the 17th century: • Tangent line problem • Velocity and acceleration problem • Minimum/maximum problem • Area problem

  4. How do you define a tangent line to a curve? Can touch at more than one point Can cross the curve

  5. Q (c+∆x,f(c+∆x)) P (c,f(c)) ∆x Slope of Tangent Line As Q approaches P, the slope of the secant line approaches the slope of the tangent line.

  6. Q (x,f(x)) P (c,f(c)) Slope of Tangent Line

  7. Slope of Tangent Line Alternate definition of derivative at a point Standard definition of derivative at a point

  8. Definition of Tangent Line The definition of a tangent line to a curve doesn’t cover a vertical tangent line. If m=∞ or -∞ or Then there is a vertical tangent line at x=0.

  9. Definition of the Derivative of a Function If you let (x,f(x)) represent an arbitrary point on the graph, the definition of the derivative of a function is

  10. Derivative Standard form Alternate form

  11. Terminology • The process of finding a derivative is called differentiation. • A function is differentiable at x if its derivative exists at x. • A function is differentiable on (a,b) if it’s differentiable at every point in the interval. • f ′ (x) is read “f prime of x”.

  12. “the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “dee dee ecks uv eff uv ecks” “the derivative of f of x” or

  13. Note: dx does not mean d times x ! dy does not mean d times y !

  14. does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)

  15. does not mean times ! Note: (except when it is convenient to treat it that way.)

  16. In the future, all will become clear.

  17. Example Find the slope of the graph of f(x)=2x-3 at the point (2,1).

  18. Example f(x)=x3+2x Find f ′(x).

  19. Example y = 2 / t Find dy/dt.

  20. Homework 2.1 (page 103) #5-9 odd 11,13, 17-21 odd 25-29 odd (part a only)

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