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Understanding the Derivative: Definition, Examples, and Graphs

This chapter covers the concept of the derivative, emphasizing its definition and significance in calculus. The derivative of a function f(x) at a point x=a, expressed as f'(a), is defined through limits. We explore how to find derivatives using examples, including polynomial functions and square root functions. The chapter also highlights the graphical representation of derivatives and differentiable functions. Additionally, it discusses important theorems related to continuity and differentiability, showcasing practical examples of limits and cusps in functions.

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Understanding the Derivative: Definition, Examples, and Graphs

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  1. Chapter 2 Section 2 The Derivative!

  2. Definition • The derivative of a function f(x) at x = a is defined as f’(a) = limf(a+h) – f(a) • h->0 h • Given that a limit exists. • Then f is differentiable at x = a.

  3. Example! • Find the derivative of f(x) = x3 + x – 1 at x = 1 • Start with f(1 + h) – f(1)/h

  4. General Example! • Find the derivative of f(x)=x3+x-1 at some point x. (this point we don’t know)Differentiation • The derivative of f(x) to get the new function f’(x) given a limit exists. The process is called differentiation.

  5. Derivative of a sqrt function • If f(x) = √x • What do the x’s have to be? • We need to figure out how to derive a new function from this using our formula.

  6. Now to some graphing ?!? • Let’s look at some graphs of functions.

  7. More graphing!!! • Graphs of derivatives.

  8. Alternative notation • f’(x) = y’ = dy/dx = df/dx = d/dxf(x) • Where d/dx is called the differential operator • Or tells you to take the derivative of f(x)

  9. Theorem 2.1 • If f(x) is differentiable at x = a then f(x) is continuous at x = a. • EXAMPLE TIME!!!!!!!!!!!!!!!!!

  10. Show f(x) = 2 if x > 2 and 2x if x≥2 • At x = 2. • Let’s graph it! And then check our LIMITS!!!

  11. Some non differentiable exampples • See Page 171, basically if there is a discontinuity in the graph, it is not differentiable at that point. • Or a “cusp” or “Vertical Tangent” line.

  12. Approximating a derivative/velocity numerically • Use the function to evaluate the limit of the slopes of secant lines! • Use f(x) = x2√(x3 + 2) at x = 1.

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