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Understanding Derivatives: The Tangent Line Problem and Beyond in Calculus

Dive into the fundamental concepts of calculus, particularly focusing on the tangent line problem. This lesson explores the definition of the derivative, illustrated through the calculation of slopes at specific points on graphs. We'll examine how to find the equation of a tangent line, understand differentiability, and see how discontinuities can affect a function's derivative. Unravel the four major problems that drove the development of calculus and grasp these concepts that are foundational for understanding motion, optimization, and areas under curves.

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Understanding Derivatives: The Tangent Line Problem and Beyond in Calculus

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  1. 2.1 The Derivative and the Tangent Line Problem I’m going nuts over derivatives!!!

  2. Calculus grew out of 4 major problems that European mathematicians were working on in the seventeenth century. 1. The tangent line problem 2. The velocity and acceleration problem 3. The minimum and maximum problem 4. The area problem

  3. The tangent line problem secant line (c, f(c)) (c, f(c)) is the point of tangency and f(c+ ) – f(c) x is a second point on the graph of f.

  4. The slope between these two points is Definition of Tangent Line with Slope m

  5. Find the slope of the graph of f(x) = x2 +1 at the point (-1,2). Then, find the equation of the tangent line. (-1,2)

  6. f(x) = x2 + 1 Therefore, the slope at any point (x, f(x)) is given by m = 2x What is the slope at the point (-1,2)? m = -2 The equation of the tangent line is y – 2 = -2(x + 1)

  7. The limit used to define the slope of a tangent line is also used to define one of the two funda- mental operations of calculus --- differentiation Definition of the Derivative of a Function f’(x) is read “f prime of x” Other notations besides f’(x) include:

  8. Find f’(x) for f(x) = and use the result to find the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)? 1

  9. Therefore, at the point (1,1), the slope is ½, and at the point (4,2), the slope is ¼. What happens at the point (0,0)? The slope is undefined, since it produces division by zero. 1 2 3 4

  10. Find the derivative with respect to t for the function

  11. Theorem 3.1 Alternate Form of the Derivative The derivative of f at x = c is given by (x, f(x)) (c, f(c)) c x

  12. Derivative from the left and from the right. Example of a point that is not differentiable. is continuous at x = 2 but let’s look at it’s one sided limits. -1 1

  13. The 1-sided limits are not equal. , x is not differentiable at x = 2. Also, the graph of f does not have a tangent line at the point (2, 0). A function is not differentiable at a point at which its graph has a sharp turn or a vertical tangent line(y = x1/3 or y = absolute value of x). Differentiability can also be destroyed by a discontinuity ( y = the greatest integer of x).

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