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# Week 4 Lecture Notes for Math 127 MATH127

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This 4 page Class Notes was uploaded by Monica Weisenbach on Friday February 20, 2015. The Class Notes belongs to MATH127 at University of Massachusetts taught by Thurlow Cook in Spring2015. Since its upload, it has received 58 views. For similar materials see Calculus I for Life and Social Science Majors in Mathematics (M) at University of Massachusetts.

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Date Created: 02/20/15

Week 4 Linear Approximation How can we find the value of x42 with linear approximation V42 is close to 4 and so we can use the tangent line derivative of x4 to get the value of x42 YtangentmhfX so m yfxh m is also the derivative The derivative of yxx 12x12 so mf 412412121414 Going back to our equation YtangentmhfX plug in the values Ytangent144242 14 is from the m calculation h is the distance between the points and fx is from the original equation yxX Solving that we get an estimation of y205 when x42 Going to the original yx42 2049 so very close Lad x 1 Xqh Continuity To be continuous at a value means the graph is not broken at that point An example would be fx x24x2 this graph has a hole at x2 no dividing by zero L L Ca lnu ouS N 0139 Cu lm ouS Divide by Zero Note Why exactly can t you divide by zero Division is ab quotient so a bquotient Taking that 10 q should be equivalent to 1 qO and that is impossible What about 00 Presumably that should work but that works out to O 0q which works for all numbers thereby violating the rule that division should only have one answer Second Derivative Basically the second derivative is looking at the derivative of a derivative f x is the derivative of fx and f x is treating f x as the original function From by looking at the f x and the f x we can tell if it s curving up or down Powers of X Remember again x nxn391 but how do we solve that for X fxxx so what is f x r x Xh vrx Xh vrx xhsv evleerh54v evlee4qx h 1 h Xh Vx hxXhX hxxhx XhX Now if you take the limit hgtgtO 1Vx0V0 12Vx 121Vx 12x12 Exponential Function and Logarithm Derivative Exponential functions are constant a variable examples are 20X and 5X What is f x for fx 2X lim fx 2Xh 2X 2X2h 2X 2X 2h1 so the f x 2X lim 2h1 2X ln2 hgtgtO h h h hgtgtO h Test h001 and the answer of 20011001 6933875 gt very close to ln2 Basically the derivative of exponential NX f x NXlnN Now that we ve done that is there a value of N that makes the derivative of an exponential function just the exponential function If fx eX then the derivative is eXIne but In means natural logarithm which means base e so lne1 So that translates to eXIne eX1 eX Also the derivative of lnX 1X note that 1X can t be gotten as a derivative from a power function 1X would be x1 meaning that it would have to come from x0 which is 1 Manipulation of Derivatives f9X fX JX SO f9 X f X 9 X Proof fgXhfgX fXhgXhfX9X fXhfXgXh9X h h h And that last is the combined equation of f x and g x afx af x afxh afx afxhfx a fxhfx af x h h 1 h These two properties make it easier to work out the derivatives of some equations Ex y 2x29x1 Take the derivative of each term on its own and add them 2x2 22x 9x 9x0 10 the derivative of a constant is zero so f x4x9 EX y 4X33X25X22 4X3 43X2 3X2 32X 5X 5XO 22 0 so f x12x26X5

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