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Unformatted text preview: MATH 2400 LECTURE NOTES: DIFFERENTIATION PETE L. CLARK Contents 1. Differentiability Versus Continuity 1 2. Differentiation Rules 3 3. Optimization 7 3.1. Intervals and interior points 7 3.2. Functions increasing or decreasing at a point 7 3.3. Extreme Values 8 3.4. Local Extrema and a Procedure for Optimization 10 3.5. Remarks on finding roots of f 12 4. The Mean Value Theorem 12 4.1. Statement of the Mean Value Theorem 12 4.2. Proof of the Mean Value Theorem 13 5. Monotone Functions 14 5.1. The Monotone Function Theorems 14 5.2. The First Derivative Test 17 5.3. The Second Derivative Test 17 5.4. Sign analysis and graphing 18 5.5. Continuity of Monotone Functions 20 5.6. A theorem of Spivak 21 6. Inverse Functions 22 6.1. Review of inverse functions 22 6.2. The Interval Image Theorem 23 6.3. Monotone Functions and Invertibility 24 6.4. Inverses of Continuous Functions 25 6.5. Inverses of Differentiable Functions 26 6.6. Some important inverse functions and their derivatives 27 References 27 1. Differentiability Versus Continuity Recall that a function f : D R R is differentiable at a D if lim h f ( x + h ) f ( x ) h exists, and when this limit exists it is called the derivative f ( a ) of f at a . More- over, the tangent line to y = f ( x ) at f ( a ) exists if f is differentiable at a and is the unique line passing through the point ( a,f ( a )) with slope f ( a ). 1 2 PETE L. CLARK Note that an equivalent definition of the derivative at a is lim x a f ( x ) f ( a ) x a . One can see this by going to the - definition of a limit and making the substitu- tion h = x a : then 0 < | h | < < | x a | < . Theorem 1. Let f : D R R be a function, and let a D . If f is differentiable at a , then f is continuous at a . Proof. We have lim x a f ( x ) f ( a ) = lim x a f ( x ) f ( a ) x a ( x a ) = ( lim x a f ( x ) f ( a ) x a ) ( lim x a x a ) = f ( a ) 0 = 0 . Thus 0 = lim x a ( f ( x ) f ( a )) = (lim x a f ( x )) f ( a ) , so lim x a f ( x ) = f ( a ) . Remark about linear continuity ... The converse of Theorem 1 is far from being true: a function f which is con- tinuous at a need not be differentiable at a . An easy example of this is f ( x ) = | x | at a = 0. But in fact the situation is even worse: a function f : R R can be continuous everywhere yet still fail to be differentiable at many points. One way of introducing points of non-differentiability while preserving continuity is to take the absolute value of a differentiable function. Theorem 2. Let f : D R R be continuous at a D ....
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