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by: Bernardo Kunze


Bernardo Kunze
Texas State
GPA 3.57

K. MacInnis

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K. MacInnis
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This 8 page Class Notes was uploaded by Bernardo Kunze on Wednesday September 23, 2015. The Class Notes belongs to MATH 1329 at Texas State University taught by K. MacInnis in Fall. Since its upload, it has received 19 views. For similar materials see /class/212750/math-1329-texas-state-university in Mathematics (M) at Texas State University.

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Date Created: 09/23/15
1329 Calculus outline 1 The difference quotient and average rates of change a Rates of change oflinear functions everyone has a constant rate of change slope b Vertical lines are not a function 1 Difference quotient w 1 Where h is the change in X c Example Find the difference quotient for linear function fX2X5 1 Work fXh 2Xh5 9 2X2h5 ii fXhfX 2X2h52X5 9 2hh 2 d Average rates of change of nonlinear functions 1 Nonlinear functions do not have a constant rate of change the rate of change caries depending where you are on the graph It is defined to be the line between two points on the graph of the function whose Xcoordinates are the end points of the interval is the slope of the line 2 Common types of functions a Constant continuous everywhere b Linear continuous everywhere c Quadratic function continuous everywhere d Polynomial continuous everywhere 6 Rational continuous everywhere except for values that make the denominator zero 3 Limits and continuity a Limits are the way of describing the behavior ofa function when the X coordinates get close to a particular value b The limit of fX as X approaches c from the left is the value that the y coordinates get close to as the Xcoordinates to the left of c get close to C i Precise definition of continuity function fis continuous at point c if the limit of fX as X approaches c eXists and if it equals fc 4 Limits of rational functions and sign charts a F ST Rational functions continuous everywhere except for the values that cause the denominator to be zero When we substitute c into a rational function there are three possibilities i fc is defined non zero denominator then the limit eXist and is equal to fc we get no where n is a nonzero number then there is a vertical asymptote at XC and the limit does not eXist ii39 We get 00 which is called an indeterminate limit and we have to do more work When we have an indeterminate limit we must rewrite the fraction as an equation fraction to determine whether the limit eXists A sign chart is just a way of depicting where a function is positive or negative Example fX XZ ZXls Find the partition points discontinuing zeros i X2 2X150 9 X3 XS 0 X3 and X5 We then should test pints on the sign chart to see where the negative and positives are 5 Introduction to the derivative a The derivative of a function is use to find the instantaneous rate of change of the function Average rate of change slope ofa secant line The derivative is the limit of the difference quotient as h approaches The line tangent to the graph offat XC is the line that passes through the pint cfc and has slope equal to the instantaneous rate of change offat c m f c d Example find the derivative of fx x25x7 i fxh xh25xh7 x22xhh25x5h7x25x7 2xhh25h h Now we still have to set the limit as h approaches 0 2x5 6 fx 2x5 a fxh 2xh5 2x2h5 2x5 h 12x5 To differentiate means to find the derivative 7 Basic differentiation rules 999691 H The derivative of any constant function is zero if fx k for some constant k then f x 0 iffis a linear function fX mX b then f39X m If fx x 1 then f39x nxn39l 11 must be a This rule Is called the power rule Example fx x5 f x 5x4 fx x fx xvz then f x 12 fx 1 2x Constant multiple rule gx 1 9Jx3 x39rso g 39 x 39 x 129 3 9 9X12 Find where the tangent line is horizontal for fx x28x10 f x 2x8 which is the slope so plug 3 in f 3 238 2 now use y1y2mX1X2 y52x3y 2x1 8 Differentials and differential approximation a U C d We can denote the change in x by Ax And the corresponding change in x for a function by yA Example find Ay for yx22x7 ifx1 at Ax3 i First find yvalue Y1 12217 6 y2 22227 15 So yA 12 3 dx f39Xl dy f39Xo dX 9 Marginal analysis a The marginal revenue function is the derivative of the revenue function So ifrx 2x25 the marginal revenue function is r x 4x b Nondifferentiable is the point where the limit does not exist 10 Concavity a Concavity is a way to describe a curve b He second derivative tells us about concavity Ifits positive then the function is concave up if its negative its concave down c The second derivative allows us to find the point of diminishing return d Currently a store sells 200 tvmonth If store invests x thousand advertising sales will be nx 3A 25X4 200 0 S XS 9 1139X 922 Xquot 11quotX 18X 3x218x3x203x6x0x0x6 This means the rate if change of sales is increasing between 0 and 6 when spending between 0 and 6000 on advertising The rate of change of the sales in decreasing between 6 and 9 spending 6000 9000 on advertising 11 Causes ofnon differentiability a Sharp bend b Vertical tangent line c Discontinuity 12 Absolute extrema and optimization a These are the lowest or highest points on a graph or a specified interval y2AJ 5 b Example on 33 So 4x0 x0 Solve for X 3 0 and 3 Abs min of 0 at x0 Abs max of 23 at x33 13 Exponential and logarithmic functions a An exponential function is a function in the form y Iquot where b is greater than 0 and 1 b The domain of any exponential function is all real numbers c The range of any exponential function is all positive numbers d A base for exponential functions that often occurs in nature is 1 e 1 X1Xz 271828183 Euler s number X gt 0 e Natural exponential function yeX 11 10g2 32 f Examples 2quot 32 overview the logarithm rules 11 5 14 Derivative of exponential and logarithmic functions lnX 10gb Xi d d 1 lnb a 7equotequot7ln i X X X ax 6x0 1 X6equot2X lnX2 b Examples 2 fX6equot2 i X c Other formulas you need to know 039 1 t 1 7 1 7 dX g X x1111 I IXab X b 1quot X ab ba39 then then fXjabba 10X bf tb39 b2 15 Function composition and chain rule a Function composition is the act of using one function as the input for another function M v b For example 11X X3 2X 3 mm W 2X 3 c The chain rule to take the derivative of a function involving composition we must use the chain rule y16X3 u 6X3 1 y 6X 32 1 d Example l6X 32 06 dX 2 dy 6 dX 2396X 3 3 6X 3 e More formulas to know E Equot equot o u d idX fu f H Li d 1 i lnu 70u39 dX u f 101quot nu Ho Li dX 16 Antiderivatives a Ifa function fI the derivative of another function F the F is the antiderivative of f IA dXLAlHlC b n1 c The indefinite integral offis written I fXdX 1 J idX 1n X c d J 8XQ X8 c i J idX 5 1n X c X e I 6XdX ex 17 Integrate by u substitution a To find the derivative of a function like y3x75we must use the chain rule J 6 X 12dX L1 6 X 1 b J L12 01 du 6 1 7 L1sz 61 i6 X 13 c 15 18 Area under a curve and the definite integral a This is about the definite integral b Id the area above the X axis is positive the area below it will be negative c The area under the cure between the X values a and b is given by the definite integral sign 391 fXdX d To approximate areas under cures we can find he areas of rectangles on some subintervals We can calculate are of each rectangle add them get the Riemann sum which gives this area 17 IO max Rb 6 7 l7 1 ZXdX 49 L max FUD Ha l 1 f 5 5 5 f Example 255 5 1953125 625 1952500 19 Average value ofa function CX a Recall that the cost per unit is EX X 1 b c Example the total cost of producing x fridges is cx10x80 b IbfXdXAverage value of a function on the interval ab a a Find the average cost per fridge at a production level of 1000 fridges cX 10X80 1080 X X X 7 80 C 1000 10718 c 1000 20 Area between curves a Area between two functions Suppose IbfXdX IbgXdX i Which can also be I fX gXdX 4 1 4 IO ZX 5dX IO A 4XdX yiX5 04AJ 375X5dx 4 ii Example y xz4x so AJ l875ASX 3 0SXS4 2113 iii Remember when we don t have a function above the other for the entire interval we have to break it into two intervals


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