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# Class Note for MATH 1314 with Professor Gross at UH

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

M1314 Lesson 3 1 Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus the tangent line question We are interested in finding the slope of the tangent line at a SpeCIfIC pomt 1B to t t Tt tanaQn i quotn quotT1 POSI39E VV slop V T1 95W 5 L We need a way to find the slope of the tangent line analytically for every problem that will be exact every time We can draw a secant line across the curve then take the coordinates of the two points on the curve Pand Q and use the slope formula to approximate the slope of the tangent line M1314 Lesson 3 2 Consider this function Sac a L 1quot e G Now suppose we move point choser to point P When we do this we ll get a better approximation of the slope of the tangent line Thanjen Sine M1314 Lesson 3 3 Suppose we move point Qeven closer to point P We get an even better approximation We are letting the distance between Pand Qget smaller and smaller What does this sound like Pand QH 5 mos 0 Une J a 05mm Ion P Q 004 tine I bonaen t Q l on 6 Now let s give these two points names We ll express them as ordered pairs 41le xhj M1314 Lesson 3 4 Now we39ll apply the slope formula to these two points JEOHYD 4 x7 SEQ HO w Cgtlt3 mm X h This expression is called a difference quotient The last thing that we want to do is to let the distance between Pand Qget arbitrarily small so we39ll take a limit This gives us the de nition of the slope of the tangent line Definition The slope of the tangent line to the graph of fat the point Px fxis given by m fx h fx h gt0 h provided the limit exists The difference quotient gives us the average rate of change We find the instantaneous rate of change when we take the limit of the difference quotient The derivative of fwith respect to xis the function f39 read fprimequot defined by f39xll1ingw The domain of f39 x is the set of all xfor which the limit exists We can use the derivative of a function to solve many types of problems But first we need a method for finding the derivative M1314 Lesson 3 5 The FourStep Process for Finding the Derivative Now that we know what the derivative is we need to be able to find the derivative of a function We39ll use an algebraic process to do so We39ll use a FourStep Process to find the derivative The steps are as follows 1 Find fxh 2 Find fx h fx fx h fx h 4 Find the limit of the difference quotient as h gets close to o h gt0 h 3 Form the difference quotient fx h fx So f39xll1ir1 h Example 1 Find the rule for the slope of the tangent line for the function fx 3x 2 Step1 FCxgtrh7 3gtltgtQ2 3x31n2 stepz gtltD gtO 903 vi h M z 3h Step3 ltX D Cgtlt 2 2L 3 h h WU 90 Step4 hm S X h 3 hao M1314 Lesson 3 6 Example 2 Use the fourstep process to find the derivative of fx x2 Step 1 ixh3 KMDZ PHYOGH h xLxhx lo in z X1 th hz HH Step2 FQGi39D QQXB in 7 th y7v 7 MC h Step3 Fgtltmx3 JXYWHn Zx h h M 39 AX Hn Step4 M JUN h 391 ozX tO 2X h39 o M1314 Lesson 3 7 Example 3 Find the derivative of fx x L FCgtlty SEgt0 X q X quot XCX4Q KCx HQ lt39 1b quot 3 xCxHD in Axxua FW aw 1 y l d C gt chrb W 15 flth3 x X I IIM J L h xCx X hr a M1314 Lesson 3 8 Example 4 Find the derivative if fx 4x2 2x 3 Q 40pm LiCX WDEZCxH O 3 7 quot Xzthh 7ax zh 3 q xhxxv NW ZXZh 3 Q Fgtlt D Fxgt M3X usz zx Zh B39M LM BXMLHJZZ14 9ltgtlth7 ux 8xh ihl zh 3X Lwa h T Q h m gt410 HEY hm 8x LHq Z 8 x 4o 2 h o h h zo lt3 x2 M1314 Lesson 3 9 Example 5 Suppose fx x2 2x5 Find the I m of fover the interval 2 5 mzafm HsJ Ccz 2ES L325 77 5 2 39 3 z i ltSgt 531534 I 23 10 20 27 lt27 zczvs q qss l From this lesson you should be able to Explain what a derivative is State the limit definition of the derivative Use the fourstep process to find the derivative of a polynomial function Find the average rate of change

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