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Understanding Derivatives: Exploring Tangent Lines and Slopes

This guide will help you grasp the concept of derivatives, focusing on finding the slope of tangent lines at specific points. You'll learn about the relationship between a function and its derivative, recognizing when a function may not be differentiable. The derivative formula is crucial for determining tangent line slopes, instantaneous speed, and object velocity. Various notations for derivatives will be explained, and examples provided to enhance your comprehension. Understanding when a function is not differentiable, such as at cusps, vertical asymptotes, or jumps, will also be covered.

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Understanding Derivatives: Exploring Tangent Lines and Slopes

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Presentation Transcript

  1. Warm Up • Find the slope of the tangent line to at x=2. • Answer: m= -4

  2. Derivative of a Function 3.1

  3. Goal • I will be able understand the relationship between a function and its derivative as well as recognize when a function will not be differentiable. • New calendar

  4. Definition • The derivative is the formula for slope of a tangent line, instantaneous speed, or velocity of an object.

  5. Formula

  6. Alternate Formula • If asked to find the derivative at a point x=a.

  7. Notation • There are many ways to denote the derivative. They can all be found at the top of page 101. • I will also give them to you now…

  8. “the derivative of f with respect to x” Y prime “the derivative of y with respect to x” “the derivative of f with respect to x” “the derivative of f of x”

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

  10. 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.)

  11. Example • Use the definition to find the derivative of at a=1. • Answer:

  12. Graphing f’(x) • To graph the derivative, estimate the slope at a few points, then plot those values on the new graph.

  13. Example

  14. When can you not find the derivative? • Differentiability implies continuity! If a function is not continuous, it is not differentiable. • Cusps/points, vertical asymptotes, jumps/gaps

  15. Homework

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