Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y). No work needs to be shown.) Explain why fx) = x 5 + x has an inverse function. c) For y = fx) = x 5 + x, use your answer to part a) to find f 1 ) 3). Note that f1) = 3. ) 10 Points) Let y = x x for x > 0. a) Find dy. Find the equation of the tangent line to y = x x at the point x = 1.
Math 6 Calculus Spring 016 Practice Exam 1 3) 5 Points) Use the fact that 1 + 1 x = e to find 1 + x x) 1 ) n. n 3n 4) 15 Points) Find each of these derivatives. a) d sin 1 x 3 ) d 1 3tan x) c) d log x 4 + x + 1)
Math 6 Calculus Spring 016 Practice Exam 1 5) 15 Points) Find each of these integrals. a) 1 + 9x e e xlnx)) 4 c) π/4 0 sinx) e cosx)
Math 6 Calculus Spring 016 Practice Exam 1 6) 0 Points) Evaluate the following its. If you use L Hospital s Rule, show where you use it and explain what type of it you are using it on. a) sinx 3 ) sin3x 3 ) x lnx) + c) 1 + x + x ) 1/x.
Math 6 Calculus Spring 016 Practice Exam 1 7) 10 Points) In the year 1980 your parents invested $10, 000 in a special bank account which earned interest compounded continuously. In the year 000 that account was worth $160, 000. Assuming no withdrawals, and the same interest rate, what will that account be worth in the year 00? 8) 15 Points) Evaluate the following integrals. a) 5 Points) x cosx) 10 Points) e x cosx)
Math 6 Calculus Spring 016 Practice Exam 1 Solutions 1) 10 Points) a) points) Write down the formula relating the derivatives f x) and f 1 ) y). No work needs to be shown.) The relationship between them is f 1 ) y) = 1 f x). 4 points) Explain why fx) = x 5 + x has an inverse function. Since fx) = x 5 + x, f x) = 5x 4 + > 0 since 5x 4 0, so fx) is an increasing function, which is then one-to-one, so it has an inverse. c) 4 points) For y = fx) = x 5 + x, use your answer to part a) to find f 1 ) 3). Note that f1) = 3. From part a) we know that f 1 ) 3) = 1 f 1) = 1 51 4 ) + = 1 7. ) 10 Points) Let y = x x for x > 0. a) dy = d ex lnx) = e x lnx) d x lnx)) = xx lnx) + x 1 x ) = xx lnx) + 1). The equation of the tangent line to y = x x at the point x = 1 is y y 0 = sx x 0 ) where x 0 = 1 and y 0 = x x 0 0 = 11 = 1 and the slope of the tangent line s = 1 1 ln1) + 1) = 1 is the value of the derivative at x = 1 from part a). So the equation we want is y 1 = 1x 1), that is, y 1 = x 1 which simplifies to y = x. 3) 5 Points) Use the fact that 1 + 1 x = e to find 1 + x x) 1 ) n. n 3n Let x = 3n so x/3 = n. Since x as n, the it equals 1 + 1 ) n = 1 + 1 x/3 = n 3n x x) x 1 + 1 x) x ) 1/3 = e 1/3.
Math 6 Calculus Spring 016 Practice Exam 1 Solutions 4) 15 Points) Find each of these derivatives. a) d sin 1 x 3 ) = 3x 1 x3 ) = 3x 1 x 6. d 1 3tan x) = 3 tan 1 x) 1 1 + x ln3) c) d log x 4 + x + 1) = 4x 3 + x x 4 + x + 1) ln) 5) 15 Points) Find each of these integrals. a) 1 + 9x Using the substitution u = 3x so du = 3, we get du/3 1 + u = 1 3 tan 1 u) + C = 1 3 tan 1 3x) + C. e e xlnx)) 4 Using substitution u = lnx) we get du = x u = lne) = 1 to u = lne ) =, so e e and the bounds of the integral are from ] xlnx)) 4 = u 4 du = u 3 = 1 1 1 3 1 3 3 1 ) 1 3 = 7 4. c) π/4 0 sinx) e cosx) Using substitution u = cosx) we get du = sinx) and the bounds of the integral are from u = cos0) = 1 to u = cosπ/4) = 1/, so 1/ ] u=1/ e u du = e u = e 1/ e 1 ) = e e 1/. 1 u=1 π/4 0 sinx) e cosx) =
Math 6 Calculus Spring 016 Practice Exam 1 Solutions 6) 0 Points) Evaluate the following its. If you use L Hospital s Rule, show where you use it and explain what type of it you are using it on. a) sinx 3 ) sin3x 3 ) 7 points) By L Hospital s Rule for a 0 0-type indeterminate form, sinx 3 ) sin3x 3 ) L H = 6x cosx 3 ) 9x cos3x 3 ) = 6)1) 9)1) = 3. x lnx) + 6 points) x lnx) is an indeterminate form of the type 0) ), which we + make into a lnx) L H 1/x x 3/ -type by writing it as + x 1/ = + 1 = = x 3/ + x x1/ = 0 + c) 1 + x + x ) 1/x. 7 points) Let y = 1 + x + x ) 1/x ln1 + x + x ) so lny) =. Then x ln1 + x + x ) lny) = x is a type 0 0 indeterminate form. L Hospital s Rule gives ln1 + x + x ) x L H = 1+4x 1+x+x 1 = 1 so y e 1 = e is the it we seek.
Math 6 Calculus Spring 016 Practice Exam 1 Solutions 7) 10 Points) In the year 1980 your parents invested $10, 000 in a special bank account which earned interest compounded continuously. In the year 000 that account was worth $160, 000. Assuming no withdrawals, and the same interest rate, what will that account be worth in the year 00? For continuous compounding, the value of the account is P t) = 10000e It where I is the annual interest rate. Then we know 160000 = 10000e 0I so 16 = e 0I gives I = ln16) 0. Then in the year 00, when t = 40, the value of the account would be P 40) = 10000e 40I = 10000e ln16) = 1000016) = 1000056) =, 560, 000 dollars. 8) 15 Points) Evaluate the following integrals. a) 5 Points) x cosx) Using integration by parts with u = x and dv = cosx), we have du = and v = sinx), so x cosx) = x sinx) sinx) = x sinx) + cosx) + C 10 Points) e x cosx) Use integration by parts twice, the first time with u = e x and dv = cosx), so that du = e x and v = sinx), giving e x cosx) = e x sinx) e x sinx) For the second integration by parts on the remaining integral, use u = e x and dv = sinx), so that du = e x and v = cosx), giving [ ] e x cosx) = e x sinx) e x cosx) cosx)e x ) = e x sinx) + e x cosx) 4 e x cosx) Bringing the last term to the other side of the equation, and then dividing by 5 gives the answer e x cosx) = 1 [ ] e x sinx) + e x cosx) + C 5