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Calculus 1 Lecture Notes Section 2.2 Page 1 of 5 Section 2.2: The Derivative Big Idea: The slope of the tangent line to a function at a point is called the derivative of the function at that point. Big Skill: You should be able to compute the derivative of a function at a point or the derivative function in general by evaluating the difference quotient limit (i.e., the limit of the secant line slopes). Definition 2.1: The Derivative at a Point The derivative of the function f(x) at x = a is defined as f a h f a f ' a lim , h 0 h provided the limit exists. If the limit exists, we say that f is differentiable at x = a. Alternative Definition of the Derivative at a Point f b f a f ' a lim b a ba Practice: 1. Find f (a) for f(x) = 4x – 9 at a = 3 using both definitions of the derivative. 2. Find the derivative of f(x) = 3x2 – 2x + 1 at x = -2 using both definitions of the derivative. Calculus 1 Lecture Notes Section 2.2 Page 2 of 5 Definition 2.2: The Derivative Function The derivative of the function f(x) is defined as f x h f x f ' x lim , h 0 h provided the limit exists. The process of computing a derivative is called differentiation. Notice that differentiation produces a function whose output is the derivative of the function f(x) at any value of x. Practice: 3. Find f (x) for f(x) = 4x – 9. 4. Find f (x) for f(x) = 3x2 – 2x + 1. 5. Find f (x) for f(x) = 2x3 + 4x2 + x + 1. Calculus 1 Lecture Notes Section 2.2 Page 3 of 5 1 6. Find f (x) for f x . x 3 7. Find f (x) for f x . x2 8. Find f (x) for f x x . 9. Find f (x) for f x 3 x 1 . Calculus 1 Lecture Notes Section 2.2 Page 4 of 5 Skill: Sketching graphs of f(x) or f (x) given the other function. Things to notice: Extrema of a graph have a slope / derivative of zero. Increasing regions of the graph have a positive derivative. Decreasing regions of the graph have a negative derivative. Practice: 10. Sketch the derivative of the function graphed below. 11. Sketch the function whose derivative is graphed below. Alternative derivative notations: If y = f(x), then we also write the derivative of f(x) as: dy df d f x y f x dx dx dx Calculus 1 Lecture Notes Section 2.2 Page 5 of 5 Theorem 2.1: Relationship Between Differentiability and Continuity. If f(x) is differentiable at x = a, then f(x) is continuous at x = a. Proof: Continuity at x = a requires lim f x f a . This is what we have to show, starting from the given x a that f(x) is differentiable f x f a lim f x f a lim x a x a x a xa f x f a lim lim x a x a xa x a f 'a 0 lim f x f a 0 x a lim f x lim f a 0 x a x a lim f x f a x a QED. This theorem captures the idea that for a derivative to exist, both one-sided limits used in the derivative must exist. Examples where the one-sided derivatives are different are functions that have sharp corners, jump discontinuities, vertical asymptotes, or cusps. The derivative also doesn’t exist where a function has a vertical tangent line, even though the function is still continuous at that point. Practice: Find the equation of the tangent line to the following curves at the point specified.: 2 x if x 0 12. Find f (0) for f x . 3x if x 0 1 13. Find f (0) for f x . x2

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lecture notes, lecture notes section, Section 1.5, Section 2, Section 1, Donald G. Luttermoser, Fall 2005, course materials, Section 4, Part B

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posted: | 1/1/2011 |

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