MATH 1451 Chapter Notes - Chapter 3: Touch 'N Go, Sport Utility Vehicle, Fu Jia

My Not EXAMPLE 1 Find an equation of the tangent line to the function y 4x4 at the point P(1, 4). SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty observe that we can compute an approximation to m by choosing a nearby point Qix, ax) on the graph (as in the figure) and computing the slope mpo of the secant line PO. [A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] is that we know only one point, P, on t, whereas we need two points to compute the slope. But we choose x# 1 so that Q* P. Then, 4x44 x-1 PQ For instance, for the point Q(1.5, 20.25) we have .5 The tables below show the values of mpo for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer mpo is to tangent line t should be m = This suggests that the slope of the 2 60 04 1.5 32.5 57.5 1.1 18.5649 13.756 1.01 |16.242 99 15.762 1.001 16.024 999 15.976 we say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this

(a) At what value of z will the curve be at its highes (b) Graph this function with a graphing utility to Find the derivatives of the functions in Problems 1-34. 3、 f(x) = ex-x" 5. gx) 500(1-01x) point verify your answer. given by 6. h(x) - 750e04 38. (a) Find the mode of the normal distribution 10, y=1-2e 12, y=e 14, y=e, + elnx 16, y = 2 18, y= 2x + 2 20, p=4qe 22, y = 4(ex)3-4 24, y = ln (c" + 1) 26, y-er' In (4 ) 28. y =- 2x 9, y= 6e 11, y=2e(x1+1), (b) What is the mean of this normal distribution? (c) Use a graphing utility to verify your answer. 15, y=e-1/x 17, y = e-1/x' + e-r 19, s = 12e, 21, y=er_ (ex)' 23. y In (e + 2) 25, y= e-kin (2x) In Problems 39-42, find any relative maxima and minima. Use a graphing utility to check your results. 40, y =- 41 y = x-ex APPLICATIONS 43. Future value If SP is invested fornyearsat 10% 42, y=ex 1 + e ix 29, y = (r" + 4)10 30, y = 32.y-3 34, y = 52-1 compounded continuously, the future value that after n years is given by the function 33, y=4" 35. (a) What is the slope of the line tangent to y xe "at (b) Write the equation of the line tangent to the graph 0.1n S -PeIn (a) At what rate is the future value growing at any ofy = xe-x at x = 1. What is the slope of the line tangent to y = e (1 + e") at x = o? (for any nonnegative n)? (b) At what rate is the future value growing after 36. (a) (e) Is the rate of growth of the future value after (b) Write the equation of the line tangent to the graph greater than 10%? why? is invested at 9%, compounded continuously, is where t is the number of years. of y-r-x/(1 x) at x = 0. 44. Future value The future value that accrues when + e 37. The equation for the standard normal probability dis- tribution is S(t) 700e 09 The mode occurs at the highest point on norm curves and equals thr

At what value of z will the curve be at its highest point? (a) (b) Graph this function with a graphing utlityto (a) Find the derivatives of the functions in Problems 1-34. 2, y=x2-3ex 4, f(x) = 4ex-In x 6. h(x) 750e004 . fx) -e 5·g(x) = 500(1-e-0.1x) 7, y=e 9, y=6e verity your answer Find the mode of the normal distribution. given by 38, 10. y = 1-2c" 2π (b) What is the mean of this normal distribution? (c) Use a graphing utility to verify your answer In x 15. y e 16, y = 2ev's /x+e In Problems 39-42, find any relative maxima and 19. ste 21, y=er_ (ex)' 23,y=ln (e4x + 2) 25, y=e-川n (2x) 27, y= 20. p4e 22, y = 4(ex)"-4er' 24. y In (e 1) 26. yeIn (4x) 28, y = minima. Use a graphing utility to check your results. 40,y= 41, y=x-e APPLICATIONS 43. Future value If $P is invested for n years at 10% 42, y=ex 29, y = (ex + 4)io 31 33, y=4" 35. (a) What is the slope of the line tangent to yxe "at compounded continuously, the future value that resul after n years is given by the function 32. y 3 (a) At what rate is the future value growing at any ih Write the equation of the line tangent to the graph ofy = xe-x at x = What is the slope of the line tangent to y=c"/(1 + e-x) atx = 0? (b) (for any nonnegative n)? (b) At what rate is the future value growing 1. after I yell 36. (a) (b) Write the equation of the line tangent to the graph (c) Is the rate of growth of the future value after 1 ofy = e-x/(1 + e-x) at x-0 greater than 10%? why? 37. The equation for the standard normal probability dis- 44. Future value The future value that accrues when is invested at 9%, compounded continuously. S(t) 700e tribution is where t is the number of years. The mode occurs at the highest point on normal curves thl and