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# Calculus I MATH 1431

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This 152 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1431 at University of Houston taught by Alexandre Caboussat in Fall. Since its upload, it has received 67 views. For similar materials see /class/208368/math-1431-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J I l Definition 0 A function f is said to take on a local maximum at c if fc 2 fx7 for all X suf ciently close to c o A function f is said to take on a local minimum at c if fc fx7 for all X suf ciently close to c The local maXima and minima are called local extreme values of I local mirzizum minimum v If f takes on a local maximum or minimum at c then either fc 07 or fc does not exist Definition The numbers 6 in the domain of the function f for with either fc 07 or fc does not exist7 are called the critical numbers of f l l First Derivative Test Suppose that c is a critical number of f 0 If f X gt O for all X lt 6 close to c and fX lt O for all X gt 6 close to c then fc is a local maximum 6 If f X lt O for all X lt 6 close to c and fX gt O for all X gt 6 close to c then fc is a local minimum 6 If f X keeps constant sign close to c fc is not a local extreme value Second Derivative Test Suppose that f c O and f c exists 0 If f c gt 0 then fc is a local minimum 0 If f c lt 0 then fc is a local maximum 0 If f c 0 the test fails I l l I Find and classify the critical numbers for the function l I Find and classify the critical numbers for the function 3 X W M 7 E l I The graph of f x is as follows Classify the critical numbers for f 44 Endpoints and Absolute Extreme Values endpoint J endpoint l endpoint i r 474 maximum Iminimum minimum E a b a domain a b domain 2 b endpoint I endpoint mimimumi maximum i 7 domain mmi domain w b Endpoints Definition If c is an endpoint of the domain of 1 then f is said to have an endpoint maximum at c if fc 2 fx for all X in the domain of f suf ciently close to c f is said to have an endpoint minimum at c if fc fx for all X in the domain of f suf ciently close to c Chap 4 What About Absolute Extreme Values Suppose f is continuous on 37 b and differentiable on 37 b 0 Compute fa and fb 0 Compute fc for every critical number c in 37 b 6 Compare these values l l Find the absolute extreme values for the function 7 1 3 1 2 fx 7 igx l EX l 6X7 3 on the interval 741 Definition A function f is said to have an absolute maximum at d if fd 2 fx for all X in the domain of f f is said to have an absolute minimum at c if fd fx for all X in the domain of f l l Find the absolute extreme values for the function fx X272le2 on the interval 712 l l Find the absolute extreme values for the function 7 1 3 1 2 fx 7 igx l EX l 6X7 3 on the interval 74700 Behavior when X gt ioo 7 fXgtOO aSXHOO fX becomes arbitrary large when X increases without bound fXgtOO aSXHOO fX becomes arbitrary large negative when X increases without bound Examples Chap 4 Behavior when X gt ioo 7 fXgtOO aSXHioo fX becomes arbitrary large when X decreases without bound fX H700 aSXH foo fX becomes arbitrary large negative when X decreases without bound Examples Polynomials fx aan an1X 1 31X 30 fxx4ix3xzxl ax 7 2N fx X3 7 sinx Recipe extended 9 Compute the value of the functions at the endpoints if any 0 Compute the behavior of the function when x a ioo if needed 6 Compute fc for every critical number 6 in 37 b 0 Compare these values l I Find the absolute extreme values for the function on the interval 17 oo l l Find the absolute extreme values for the function fx X272X1 on the interval 7007 O MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J I l Newton Method Geometrically Approximation of roots by using the tangent lines i Newton Method Formula Let f be a twice differentiable function and suppose a is a real number at which fa 0 If f a 7 O and X0 is sufficiently close to a then the iteration fx Xn1Xn Fix n0123 n will converge rapidly to the root 3 Newton method Newton Raphson method Chap Newton method can go bad 41 The Mean Value Theorem bfb a a When is the tangent line parallel to the secant line J l I How many points are there between 71 and 3 where the tangent line is parallel to the secant line l I How many points are there between 71 and 3 where the line is parallel to the secant line Mean Value Theorem If f is differentiable on the open interval 37 b and continuous on the closed interval 37 b then there is at least one number 6 in 37 b for which He rib a or equivalently fb 7 fa f cb 7 a l I Verify the mean value theorem for fx 3X7 X2 on the interval 717 Particular Case Rolle s Theorem Let f be differentiable on the open interval 37 b and continuous on the closed interval 37 b If fa O and fb 0 then there is at least one number 6 in 37 b for which fc 0 Are there values between a and b where zero derivative is equal to How many l I Are there values between a and b where the derivative is equal to zero How many But here fa fb I l Chap 4 Rolle s Theorem Generalization Let f be differentiable on the open interval 37 b and continuous on the closed interval 37 b If fa fb not necessarily equal to zero then there is at least one number 6 in 37 b for which fc 0 Example Let fx X3 7 4X on O7 3 Verify the mean value theorem Let fx X3 7 4X on 02 Verify the Rolle39s theorem 7 Show that the function fx 2X3 5X 1 has exactly one zero Example Show that the function fx 2X3 5X 71 has exactly one zero One Last Remark Let f be differentiable at X0 0 If f x0 gt 0 then fX0 7 h lt fX0 lt fX0 h7 for all positive h sufficiently small 0 If f x0 lt 0 then fX0 7 h gt fX0 gt fX0 h7 for all positive h sufficiently small Example fxx2X7 X02 MATH 1431 19328 J Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J I l Increasing and Decreasmg Func Ions consta M m chSng a swap 39quotWasmg EvErywhere Let f be differentiable on an open interval I 0 If f X gt 0 for all X in I then f increases on I o If f X lt 0 for all X in I then f decreases on I 0 If f X 0 for all X in I then f is constant on I l I Give the intervals of increase and decrease for the function 1 5 fx 7X3 i 5X2 i 6X 7 3 3 l I Give the intervals of increase and decrease for the function fx gx3 gxzi 6X7 3 Starting from f x what is the possible shape of x l I Find f given that f x 6X2 7 7X7 5 and f2 1 43 Local Extreme Values Chap 4 Can you Identify the Local Extrema Chap 4 sea 4 Can you Identify the GlobalAbsolu e Extrema 2 3 1 39u If x l 4321DK123 x4 15 I x J V2 Definition 0 A function f is said to take on a local maximum at c if fc 2 fx7 for all X suf ciently close to c o A function f is said to take on a local minimum at c if fc fx7 for all X suf ciently close to c The local maXima and minima are called local extreme values of If luca mirafxum mlmmum Any link with the derivative J If f takes on a local maximum or minimum at c then either fc 07 or fc does not exist Consequence Local maxmin have to be there Definition The numbers 6 in the domain of the function f for with either fc 07 or fc does not exist7 are called the critical numbers of f Examples I Emammum 3 T l IucaI mimmum 2 A 037 2 rm112 r I 39 I I I I I l1 I l I I I I m xi L3 l I Find the critical numbers for the function 1 1 fx 5X2 X276X2 l I Find the critical numbers for the function fx sinx l I Classify the critical numbers as a local maximum or local minimum This is called the first derivative test I First Derivative test Suppose that c is a critical number of f o If f X gt O for all X lt 6 close to c and fX lt O for all X gt 6 close to c then fc is a local maximum 0 If f X lt O for all X lt 6 close to c and fX gt O for all X gt 6 close to c then fc is a local minimum 0 If f X keeps constant sign close to c fc is not a local extreme value local minimum x A C f c dues not exist I Example Use the first derivative test to classify the local extrema of the function fx x4 7 8x3 22x2 7 24x 4 Suppose that f c O and f c exists 0 If f c gt 0 then fc is a local minimum 0 If f c lt 0 then fc is a local maximum Counterexamples l I Find and critical numbers for the function 1 1 7 X3 X2 6X73 l I Find and classify the critical numbers for the function 1 1 fx igxs i 5X2 i 6X7 3 l I Find and classify the critical numbers for the function l I Find and classify the critical numbers for the function 3 X W M 7 E l I The graph of f x is as follows Classify the critical numbers for f l I MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 6 On the last day to drop a course or withdraw with a 39W39 the front desk of the Math Department may sign drop forms ifl am not available httpwwwmathuheduNcaboussatmath1431 l I l Upper and Lower Riemarnanums 132 2 uuuu mm Luwu sum 1 m Partition of 37 b a X07X1 7Xr b Definition Upper sum UfP Mlel MQAXQ MnAXn Lower Sum LfP mlel 772sz mnAxn Definition A function f defined on an interval 3 b is integrable on 3 b if there is one and only one number I that satisfies the inequality LfP I UfP for all partitions P of 3 b The unique number I is called the definite integral or more simply the integral of f from 3 to b and is denoted by I fxdx Notations abfxdxor abftdtor abfsds I l Example We Have Seen More When F is non negative the area under the curve fx between a and b is given by MW HWHamp where FX is an anti derivative of f Therefore b a fxdx Fb 7 Fa whether f is non negative of not 0 When f is nonnegative the integral represents the area between the graph of f and the X aXis 9 When f is not nonnegative what is the signification of the integral l I Find the area bounded by the curve fx sinx and the X axis for 0 g X g 7139 AX sinx AX 7cosx C AO 0 implies C 1 and AX 17 cosx Total Area A7r 17 cos7r1771 2 I l Compute the lower and upper Riemann sums for fx sinx over 077139 with respect to the partition P 07r2737r477r l I Compute the lower and upper Riemann sums for fx sinx over 077139 with respect to the partition P 07r2737r477r l I 1 Compute X3X2X1dx 0 l I 5 Compute dx 0 53 The Fynction Fx ftdt a l I Find the area bounded by the curve fx X2 7 3X 7 4 and the X axis on the interval 727 3 l I Find the area bounded by the curve fx X2 7 3X 7 4 and the X aXis on the interval 727 3 l I 71 Compute X2 7 3X 7 4 dx 72 3 Compute X2 7 3X 7 4 dx 71 3 Compute X2 7 3X 7 4 dx 72 If f is continuous on 37 b and alt c lt b then 39ftdtCb39ftdtabftdt Area Area Total Area J When a lt b Consequence C Iftdt 0 Examples 0 Compute X2 X dx 1 0 Compute cosxdx So far we know abftdt Remember In Chapter 3 we defined Fl before defining Fx as a function of X Question Can we define the integral as a function of X The answer is yes By definition Fx ftdt Let f be a continuous functions on 37 b The function F defined by X Fx ftdt is continuous on 37 b differentiable on 37 b and has derivative FX fx7 for all X 6 37 b Examples On the interval 01 If ft 1 Fx If ft t2 t1 Fx If ft sin5t Fx l I X 7r Compute tzisint2 dx and tzisint2dt 0 0 l I MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 httpwwwmathuheduNcaboussatmath1431 J I l If f is continuous on 37 b and alt c lt b then 39ftdtCb39ftdtabftdt Area Area Total Area J When a lt b Consequence C Iftdt 0 Let f be a continuous functions on 37 b The function F defined by X Fx ftdt is continuous on 37 b differentiable on 37 b and has derivative FX fx7 for all X 6 37 b Examples On the interval 01 If ft 1 Fx If ft t2 t1 Fx If ft sin5t Fx l I X 7r Compute tzisint2 dx and tzisint2dt 0 0 54 The Fund mental Theorem of alculus l I Give a formula for the area bounded by the curve fx and the X aXis on the interval 721 l I Give a formula for the area bounded by the curve fx 2X3 7 X and the X aXis on the interval 721 l I Give a formula for the area bounded by the curve fx 2X3 7 X and the X aXis on the interval 721 l I 1 Compute 2X37XdX 72 l I b When f is not non negative fxdx represents the net area 3 area above the X aXis MINUS area below the X aXis Two types of questions 0 Compute the net area fxdx 0 Compute the real geometric area between the graph of f and the X aXis l I Region I has area 2 region II has area 3 and region I has area 2 3 What is fxdx7 73 l I Region I has area 2 region II has area 3 and region I has area 2 1 What is fxdx7 73 l I Region I has area 2 region II has area 3 and region I has area 2 1 What is fxdx7 71 l I Region I has area 2 region II has area 3 and region I has area 2 What is the area bounded by the X axis and the graph y fx J Let s Remember Let f be continuous on 3 b A function G is called an anti derivative for f on 3 b if G is continuous on 3 b and GX fx for all X E 3 b In particular Fx ftdt is an anti derivative for f The Fundamental Theorern of Calculus Let f be continuous on 37 b If G is any antiderivative for f on 37 b then bftdt Cb 7 63 Examples 4 Evaluate dex 1 Evaluate sintdt 0 4 Evaluate 537352s 1ds 1 Examples 1 2X Evaluate 0 de 1 1 Evaluate 7dr 0 t4 1 l I Function fx Antiderivative CX X frl r rational number r 7 71 sinx cosx cosx sinx sec2x tanx secx ta n X sec X csc2x 7 cotx csc X cotx 7 cscx hap 5 Linearity of the Integral fag gx dx fx dx fax dx abozfxdxozabfxdx abon x gxdxaibfxdx ibgxdx b b If fx 2 gx on 37 b7 fx dx 2 gx dx 3 2X3 3X4 2sin5x dx 1 571s 2 d5 MATH 1431 19328 J Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J I l Chap 4 4 How to Find Absolute Extreme Values Suppose f is continuous on 37 b and differentiable on 37 b 0 Compute fa and fb 0 Compute fc for every critical number c in 37 b 0 Compare these values l l Find the absolute extreme values for the function fx i x3 le 6X7 3 on the interval 74700 0 Critical numbers 0 Endpoints 0 What happens when Behavior when X gt ioo 7 fXgtOO aSXHOO fX becomes arbitrary large when X increases without bound fXgtOO aSXHOO fX becomes arbitrary large negative when X increases without bound Examples Chap 4 Behavior when X gt ioo 7 fXgtOO aSXHioo fX becomes arbitrary large when X decreases without bound fX H700 aSXH foo fX becomes arbitrary large negative when X decreases without bound Examples Polynomials fx aan an1X 1 31X 30 fxx4ix3xzxl fx 72xxQ fx 7x3 x2 100000x 71 Recipe extended Compute the value of the functions at the endpoints if any 0 Compute the behavior of the function when x a ioo if needed 6 Compute fc for every critical number 6 in 37 b 0 Compare these values l I Find the absolute extreme values for the function on the interval 17 oo l l Find the absolute extreme values for the function fx X22X1 on the interval 7007 O More Difficult Find the absolute extreme values for the function ix1 7X71 f X on the interval 27 oo Of all the rectangles with a given diagonal of length 1 show that the square has the largest area l I Of all the rectangles with a given diagonal of length 1 show that the square has the largest area 45 MinMax Problems We want to talk about optimization problems what is the best shape what is the best angle what is the best position etc i We look for the minimummaximum of some function maximize the area maximize the perimeter maximize the volume etc A triangle is formed by the coordinates axis and a line going through the point 27 5 with negative slope m Question Determine the slope of this line such that the area of the triangle is minimum Let39s try l I A triangle is formed by the coordinates axis and a line going through the point 27 5 with negative slope m Question Determine the slope of this line such that the area of the triangle is minimum Strategy 0 Draw a representative figure and locate the relevant quantities on it Strategy 0 Draw a representative figure and locate the relevant quantities on it 0 Identify the quantity to be maximizedminimized and find the formula for this quantity Strategy 0 Draw a representative figure and locate the relevant quantities on it 0 Identify the quantity to be maximizedminimized and find the formula for this quantity 0 Express the quantity to be maxmin as a function of one variable only Use the given relations to do such simplifications Strategy 0 Draw a representative figure and locate the relevant quantities on it 0 Identify the quantity to be maximizedminimized and find the formula for this quantity 0 Express the quantity to be maxmin as a function of one variable only Use the given relations to do such simplifications 0 Determine the domain of the function Strategy 0 Draw a representative figure and locate the relevant quantities on it 0 Identify the quantity to be maximizedminimized and find the formula for this quantity 0 Express the quantity to be maxmin as a function of one variable only Use the given relations to do such simplifications 0 Determine the domain of the function 6 Find the extreme values critical numbers endpoints etc l I A string is 3 feet long We want to either form a square a circle or a square and a circle from the string so that the total area is maximized How should we proceed l I A string is 3 feet long We want to either form a square a circle or a square and a circle from the string so that the total area is maximized How should we proceed

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