Second Derivatives 2 CW A. Find d2 or f\"(x) for the following functions. x [ml 1. f(x)=%x3%x2x+5 6) f(x) = ln(cos 5x) f'(x) = ( sin 5x)(5) cos 5x = _ sin 5x cos 5x @ Q) = 5 tan 5x f "(x) = 5 sec2 5x (5) - 25 sec2 5x 1 3. y=x510x3+3x8 27 ' HOMEWORK 28 Date: 1. Find the second derivative of y = 5 sin 2x + 3e-2x _ 1 X 2. Find for y = In(5 + sin x). 3. Given that y = ex , show that 1 day 4 dx 2 - 2X- dy _ 2y = 0. dx 4. Find Yify = - e 2x dx 2 1 - X 5. Given that y = 1 show that dry dy + 215 = 8x + 2x 2 x 2 + 1 dy 2 dx V2 x 2 + 1) 15 V2x2 + 1 CLONE Mar 04 6. Ify = 5xe-3x , prove that + 6 + 9y= 0. CLONE dx 2 dx Oct 07 7. Find the second derivative of y = In(sin 4x). CLONE Apr 10 8 . Find Dy ify = In (x3 - 2) . CLONE Sep 11 9. Find the second derivative of y = e2x sin x. CLONE Mar 14 10. Find - for y = - + 5x. CLONE Sep 14 28HOMEWORK 29 Date: Implicit Differentiation A. Differentiate each of the following with respect to x by using implicit differentiation. Example 1. 3y2 - sin2 4x = tan(xy) cos (y2 ) = 2xy - 10 - sin (y2 ) 2y y = (uv' + vu' ) - 0 u = 2x V= y u' = 2 v' = dy dx - 2y sin (y 2 ) ay = 2x- xay + y (2) dx dx - 2y sin (y2 ) dy = 2x- dy + 2y dx dx dx - 2y sin (y2 ) ay _ 2x ay = 2y dx - 2y sin (2 ) - 2x = 2y dy 2y = dx -2y sin (y2) -2x y = y sin (y2 ) + x 2. In (xy2) = _+ x2 3. Vy + x2 sin y = 7x - 4 y 29HOMEWORK 30 Date: 1. If yz = cos (x2 + y2 ) + exy, find dy by 11. Find y for x2 + 3xy - y3 = cos 2x using dx dx implicit differentiation. CLONE implicit differentiation. CLONE Nov 05 Mar 14 2. Find for xy2 + 2x3 = 5 sin(2x + y) -ey. 12. Find - for 2 + 5x In(3y) = eBy + cos2 x dx CLONE using implicit differentiation. CLONE Apr 06 Mar 15 dx 3. Find for In(3xy) - Vx2 + 2y = 3 by CLONE 13. Find for y2 In(2x) + sin(2y) = x2y3 by using implicit differentiation. Oct 07 dx using implicit differentiation. CLONE Mar 16 4. Find the gradient of the tangent to the curve x2(y - 3) + 3 tan 3y - 10 = 0 at the 14. Find -for In (sin y3 ) _ 2X = ezy by point (1, 0). CLONE Oct 08 using implicit differentiation. CLONE Mar 17 dx 5. Find -for 5xy = In(sin 2x + cos 2y) by 15. Given 5ye1-X + 8 cos(2y) = x3 + 7. using implicit differentiation. CLONE Apr 09 a) Find using implicit differentiation dx 6. Using the implicit differentiation, find dy b) Find the equation of the tangent line dx to the function at the point (1, 0). for 4x3 12 + exy = sec(1 - x). CLONE CLONE Apr 10 Jan 18 16. Given y2e5x + cos(xy) = 2 + In(x + 1). 7. Find for 3 sin(x + y) - 2ey = 6x3 by dx a) Find using implicit differentiation. using implicit differentiation. CLONE dx Oct 10 b) Find the gradient of the tangent line 8. Let y be a function of x. If e 5x-4y = In(x3y), for the function at the point (0, 1). CLONE find by using implicit differentiation. Dec 18 dx CLONE Sep 11 17. Given the function In (3-2el-x) + x2y + 3 9. Find for cos(5x + 5y) = x5 + 4xy by 3x CLONE dx + 2y. Jun 19 using implicit differentiation. CLONE x + 2 Mar 12 a) Find using implicit differentiation. dx 10. a) Find -if 10 - sin 3y = 15 - 2x by b) Find the equation of the tangent line dx to the function at the point (1, 2). using implicit differentiation. b) Hence, find the equation of the 18. Given e3xy - cos y = In(2x + 3). CLONE tangent line to the curve at the point Dec 19 (- 5, 0). CLONE Mar 13 dx a) Find using implicit differentiation. b) Find the equation of the tangent line to the function at the point (- 1, 0). 30Linear Approximations A. Solve each of the following. [ml 1. Use differentials to estimate the value of Use differentials to estimate the value of (5.012)34 correct to 3 decimal 5 63.8 correct to 3 decimal laces. laces. ~3/63.8 p p Let f(x) = f; X: 638 \\5 xo = 64 1 'l f(x)= x55x_5 5X =XX0 1 _l 5 -5 =63.864 ' : _ 2 _ 3 f(x) 2x +3x :_0.2 = _L + i 4 2'5 3x3 f(xO '+ 6x) = f(xo) + m0 ) 5x 5 63.8 =f64 + f' 64 O.2 v m ( ) '( ) ( ) 5 1 5 = E j + + ( 0.2) 3 4 = a . 2(_ 02) 4 768 = 6.736 3. Use differential to estimate the value of 2. Given f(x) = is + |n(2x-5). Estimate the X value of f(3.01) using differentials correct to three decimal places. 31 (3.99)3 + correct to two decimal (3.99)2 places. If f(X) = x2\\/1 + x3 , find the differential df. Hence, estimate the value of 2 3 CLONE (2.02) w/1+(2.02) . Mar\" Use differentials to estimate 2 1 (8.01)5 (8.01)5 + 7. Give your answer correct to three (3) decimal places. CLONE Nov 05 Use differentials to estimate 1 x/15.97 + to three 3 decimal 15.97 ( ) places. 1:23: Given f(x) = \\/283x2 . Use differentials to estimate the value of 2833(11)2 . Give your answer correct to .two (2) decimal places. CLoNE Apr07 Use differentials to approximate 4+|n(1.2) correct to two (2) decimal places. CLONE Apr08 Use differentials to estimate f(1.99) given _ , _ 6x that f(2) - 8and f(x) m [Hint f(XP + Ax) = f(xo) + f'(x0 ) Ax] CLONE Oct 08 Given f(x) = 2 . if x changes from 2 x 1 to 2.04, estimate the value of f(2.04) using differentials. Give your answer correct to THREE (3) decimal places. [Hint: f(xo + Ax) z f(xo) + f'(x0)Ax] CLONE Apr 09 to estimate cos [% + 0.01] correct to three (3) decimal CLONE Oct 10 Use differentials places. 32 10. 11. 12. 13. 14. 15. 16. 17. Use differentials to estimate the value of \\/24.98 and give your answer correct to three decimal places. CLONE Apr 11 Given f(x) = \\/3X2 2 . Estimate the value of f(3.01) using differentials. Give your answer correct to three (3) decimal places. CLONE Oct 12 Use differentials to estimate the value of \\/15.8 3/158 correct to three decimal places. CLONE Mar 16 Use differentials tO estimate the value of 1 {255.99 + correct to four $255.99 - CLONE decnmal places. Mar 17 Given f(x) = 12 + |n(x 2). Estimate the X value of f(3.01) using differentials. CLONE Jan 18 Use differential to estimate 4/159 1 correct to four decimal places. 415.99 CLONE .' Jun 18 Use differentials to estimate 77.99 + (7.99)2 correct to two decimal places. CLONE Dec 18 Use differentials to estimate the value of |n(0.99) + 3(0.99)5. ' CLONE Jun 19 Use differentials to estimate the value of 4 480.8 correct to four decimal J4 80.8 CLONE places. Dec 19