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# Calculus and Analytic Geometry I MATH 151

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This 17 page Class Notes was uploaded by Angelina Steuber on Monday October 19, 2015. The Class Notes belongs to MATH 151 at Radford University taught by William Case in Fall. Since its upload, it has received 19 views. For similar materials see /class/224680/math-151-radford-university in Mathematics (M) at Radford University.

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Date Created: 10/19/15

Math 151 Section 21 Limit Definition of a Derivative Rate of change slope Tangent lines Secant lines The limit definition of a derivative 6 xhfxh Find the slope of the tangent line m Y2 y1 x2x1 mhmfxhfx h xh x mhmfxhfx hgt0 h Therefore the derivative of the function is fxhfx h f x 13301 Example 1 Use the limit definition of a derivative to find the derivative of the function f x 3x 4 f x1h1g1 fxhfx h 3xh4 3x4 h 3x3h4 3x 4 h f x 1irr01 f x 13 f x f x 1133 f x 3 Example 2 Given f x 5x 3 nd f 39x using the limit definition of a derivative f x1hlig1 fxhfx h th5xh3 5x3 hgt0 h i 5x5h3 5x 3 Inf Example 3 Given f x x2 2 nd f x using the limit definition of a derivative f x 1g xh2 2 x2 2 h fxhfx h 1iIn hgt0 2 11mxhxh 2 x 2 hgt0 h i x2 xhxhh2 2 x2 2 11m hgt0 h i x2 2xhh2 2 x2 2 11m hgt0 h i 2xhh2 limi hgt0 h 11mh2xh hgt0 h lim2xh2x02x kw Example 4 Given f x x2 5 nd f 39xusing the limit definition of a derivative fxhfx f x 1g h 2 2 hmxh 5 x 5 hgt0 h 2 hmxhxh5 x 5 hgt0 h i x2xhxhh25 x2 5 11m hgt0 h i x22xhh25 x2 5 11m hgt0 h i 2xhh2 11m7 hgt0 h zhm h2xh hgt0 h lim2xh hgt0 2x0 2x Example 5 Find the derivative using the limit definition of a derivative f x x2 3x Solution fxh fx h f39x 13301 IN 12 gxh 3xh x2 3x h f hmxhxh3x3h x2 3x x xgt0 h l x2xhxhh23x3h x2 3x lim hgt0 h l x22xhh3x3h x2 3x fang 2 f39xlimwlimwlirn2xh32x032x3 hao h hao h kw Example 6 Find the derivative of f x x3 4 m z fx Ax xh3 4 x3 4 lim mo h 3 zhmxhxhxh 4 x 4 mo h l x2 2xhh2xh 4 x3 4 lim Ho h l x3 3x2 3x2hh3 4 x3 4 lim Ho h l 3x2 3x2 h3 lim hgt0 lim hgt0 1hing3x2 3m h2 3x2 3x0 02 3x2 h3x2 3m hz h Example 7 1 Find the derivative of f x 4 x f x mth hgt0 h 1 1 h lim kw ww nmltxh4gtltx 4gt x 4ltxh4gt kw h LLh 4 1mxh 4x 4 x4xh4 kw h M dinnw kw h M le kw h A le M70 h 110 kwxh 4x4 xo 4x4 1 x 42 Example 9 Find the equation of tangent line to the curve f x x2 3x at the point 1 2 x h x f 21th h A x h2 3x h x2 3x h x hx h 3x 3h x23x h x2 xhxhh2 3x 3h x2 3x h x2 2xhh 3x 3h x2 3x h 2m hz 3h h f39x 13301 f39x 11301 39 hm f X f39x1hino1 13Wh2xh 32x0 32x 3 Now nd the slope f39121 3 13 m 1 yY1mxx1 y 2 1x 1 y2 x1 y x 1 S x Z S xl xL I I 9x e Hx G Wm t jt w m e x 11x W 49 36Sx w W ZWw w w 1 9m 1 0 1 wWe xI wf w wwxv W g x x f uopoun 9111 Jo eAImAIJep 9111 puICI e x 0x e xwe HH w e xwe HH ww QX 71x 71 H III III HF I I 11 hm I edumxg vxvzx 104w I I Z I y 31 y 07 Wu e W I I gzxz11xwl zxz11xquot I 11 11 rnn zxz0x zxz11x w n El 71 0611 71 CHI 11 OH HI III HI zxz11x zxz11x zxz11x zxz11x z H x ZH z11x zx z11x zx m V m 71 ZX Z71x 51 zxzyx W xf 1xf I I 1 1 0911 Hi 36 z x x f I J0 eAIwAuep 9111 WE 01 edumxg Math 151 Section 24 Chain Rule The chain rule dl dx du dx Example 1 y 3x2 55 Let yu5 whereu3x253du6x dy dy du a g y39 51441114 y 53x2 5x4 6x y 30c3x2 5x4 Example 2 y x2 3x7 Let y u7 where u x2 3x3 du 2x3 111 dx du dx y39 7147711114 y39 71461114 y39 7x2 7x6 2x 3 y39 14x 21x2 3x6 Example 3 Find the derivative of y C2 6x yVx2 6x 1 yx2 6xgt2 1 Let y142 whereux2 6x3du2x 6 1 171 iu2 du y 2 1 39iu 2du y 2 1 y39x2 6x 22x 6 2x 6 y l 2x2 6x2 2x 6 xx2 6x Example 4 Find the derivative of f x C2 6x yVx3 3x 1 yx3 3xgtE 1 Let y142 whereux3 3x3du3x2 3 1 1 1 2 2 yu2 3114 2dux3 3x 23x2 3LL 2063 3x 2 x3 3x Example 5 y 962 43 1 Let yu3whereux243u2x 1 71 y 3u du 1 7 y 3u du du y 72 314a 2x y 2 3x245 2x y W Example 6 Find the derivative of y cos5x y cos5x u 5x du 5 y cosu y39 cosudu y39 5cos5x Example 7 Find the y 8 y e u 3x2 du 6x y39 e du y39 e3 6x y39 6xe3x2 Example 8 Find the y at ux2 6x du2x 6 y39 e du y39 at 2x 6 y39 2x 624 Example 9 Find the derivative of y cos5x y 63X cos6x r y i 63X cos6x 1cos6xe3x dx dx y39 36 cos6x 6e sinx y39 36 cos6x 2 sin6x Example 10 Find the derivative of y 4xex2 y 4xex2 d 2 d 2 7 4x ex 7 4x y dx dx y39 4e 4x2xex2 y 46X2 8x26 y39 4 8x2 ex2 The derivative of the natural logarithm function d 1 7 1 7 dx nx x Example 11 Find the derivative of f x 41nx f x 41 5 X X Chain Rule for the natural logarithm ilnuldu dl dx u u Example 12 Find the derivative of f x 1n3x3 Let fx lnu where u 3x2 3 du 6x du 6x 2 x2727 f 14 3x2 x Example13 Find the derivative of fx W sin3x f x 2 e4 sin3x sin3xe4xz f x 8x9462 sin3x 3 cos3xe4 2 fx 8x9462 sin3x 39462 cos3x Example 14 Find the derivative of y x3 1n5x y39 x3 mm 1n5xgt39x3 y39 3x21n5x x3 5x y39 3x21n5x x2 Example 15 Find the slope of tangent to the function y x2 2x3 at the point 127 y x2 2x3 Let yu3 where ux2 2x3du2x2 y39 3143711114 y39 31421114 y39 30c2 2x2 2x 2 y39 6x 6x2 3x2 Slope y39 61 612 212 6 61 2f 1232 129 108 m 108 Example 16 The equation of motion of a particle is st t3 6t where s is in meters and t is seconds a Find the velocity and acceleration as a function of t W s39t 3 6 3 6 at s t 61 b Find the velocity after 2 seconds W 3 6 3 6 2 m v232 612 6 4 7 s c Find the acceleration after 2 seconds at s t 61 a26212 2 S Example 17 The equation of motion of a particle is st 2t3 12t where s is in meters and t is seconds a Find the velocity and acceleration as a function of t W s a 321 12 6 12 at s t 12t b Find the velocity after 3 seconds 110 6t2 12 113 632 12 54 12 42 E S c Find the acceleration after 3 seconds at s t 12t a3 123 36 2 S

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