Calculus and Analytic Geometry I
Calculus and Analytic Geometry I MATH 151
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This 8 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 34 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 22 Derivatives of polynomials Review of the limit de nition of a derivative x fxx 2x fxx34 fx2x 4x Power Rule Iffx 0x then fx no Constant Rule If f x C where C is a constant then f x 0 Proof of the power rule Let fx x and nd f39x f39x 13301 xhquot x h fxh fx h lim hgt0 M nx Hh 1517th rlxh H h x Hm 1111 h Examples using the power rule Example 1 If f x 2x3 then find f 39x using the power rule fx 3 2x3 1 6x2 Example 2 Iffx x4 2x3 3x then find f39x using the power rule f39x 4x 32xH 3 4x3 6x2 3 Example 3 If f x i2 then find f 39x using the power rule x 1 f X 2 X 2 x fx 2x 2x 3 L2 6 3 Example 4 1 4 i i If f x 7 7 then find f x usmg the power rule x x x73 42 fx 3x 24x1 3x 4 8163 7 3 4 X 3 3 X Example 5 If f x J then find f 39x using the power rule 1 fx J x2 1 Ll 1 xzixz zixzzizi f 2 2 Example 6 If f x 6d then find f 39x using the power rule 1 fx 6f 6x5 L 1 L1 1 fx 6x2 3x 2 i J R mH Example 7 If f x 23V then find f 39x using the power rule 1 fx 2 2xa 1 2 fXl2x31Ex3 2 72 3 3 33x2 3 3x3 The exponential function The definition of the number 3 eh l 1 eis the number such that lim hgt0 The derivative of the exponential function fxex fxh fx h h exeh ex exeh 1 h i e e i 11rn7 11rn lim hgt0 hgt0 h h 4 exlhiehh1exaex Definition The derivative of f x exis f 39x ex f x 13301 h i e 11rnex hgt0 Example 8 Find the derivative of f x 66X f 39x 66X Example 9 Find the equation of a tangent line to the function f x x2 x through the point 12 Find the derivative of f x x2 x f39x 2x 1 Find the slope of the tangent line m f391 21 1 2 1 3 yy1 quot106961 y 2 3x 1 y 2 3x 3 y 223x 32 y3x 1 Example 10 Find the equation of a tangent line to the function f x x3 ex through the point 01 Find the derivative of f x x3 ex f39x 3x2 ex Find the slope of the tangent line m f390 302 eo 0 1 1 yy1mxX1 y 11x 0 y 1x y 11x1 yx1 Example 11 The equation ofrnotion ofa particle is st t3 74t Where s is in meters and t is seconds a Find the velocity and acceleration as a function of t vt s t 33quot 7 4 3t2 7 4 10 s t 6t b Find the velocity after 2 seconds vt 33quot 7 4 31 7 4 Va 7321 74 1274 8 E S c Find the acceleration after 2 seconds 111 s t 6t 12 7 62 12 32 S Derivatives of Trigonometric Functions Review liIn IHo sinh 7 1 h moo w mugs of 1 1 awn 1500 000500 000quotS 1 ow DUES my7soo 1 ow 10mg my7soo 7uys L 1410500 0H1 mn 1 14 wsoowuys 11 w A 11 we 4 411 m ltygtuysltxgtsoo 391 11 w A rugs 4 11 Iuys m A 7 0H1 w A q w 391 W auys J0 amman mu 0H1 quot1H 0 4 L A 14 10500 The Derivative 0f Cosine fxhfx fx 1h1ig h hm cosx h cos x hgt0 am cosx cosh sinx sinh cosx hm cosx cosh cosx hm smx smh hgt0 h hgt0 h hm cosxcosh 1 Smoohm smh hgt0 h hao h cosh 1 sinh h h cos x lim sin x lim hao hgt0 cosx0 sinx1 sinx Rule 2 If fx cosx f x sinx The Derivative 0f Tangent If f x tanx then f x sec2 x Example 12 Find the derivative of y 2x2 5 cosx y 2cscx 5 cosx d d y amz a6 cosx y39 4x 5 sinx
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