Final Exam Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the differential equation with the appropriate slope field. 1) y = x y A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) -6x cos 8x dx 1
3) e2x x2 dx Solve the differential equation. 4) dy dx = 8x 7sec y 5) dy dx = 9y 2 x Find the derivative of y with respect to x. 6) y = tan-1 8x 5 Solve the problem. 7) A certain radioactive isotope decays at a rate of 2% per 100 years. If t represents time in years and y represents the amount of the isotope left then the equation for the situation is y = y0e-0.0002t. In how many years will there be 89% of the isotope left? 8) A loaf of bread is removed from an oven at 350 F and cooled in a room whose temperature is 70 F. If the bread cools to 210 F in 20 minutes, how much longer will it take the bread to cool to 140 F. 9) The coroner arrives at the scene of a murder at 11 p.m. She takes the temperature of the body and finds it to be 89.7 C. She waits 1 hour, takes the temperature again, and finds it to be 88.4 C. She notes that the room temperature is 73 C. When was the murder committed? 2
Find the derivative of y with respect to the independent variable. 10) y = 5x 11) y = 9cos 12) y =log5 sin cos e 6 Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 13) y = xln x 2 14) 9x23x3 dx 1 15) dx x 5 + 8 ln x /20 16) (1 + etan 5x) sec2 5x dx 0 17) dx x 3 + 8 ln x 3
Find the derivative of y with respect to x, t, or, as appropriate. 18) y = ln (cos (ln )) Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 19) y = 2x + 3, y = 0, x = 0, x = 1 20) y = x2, y = 16, x = 0 21) y = x2 + 5, y = 4x + 5 Find the area of the shaded region. 22) 23) sin t (7 + cos t)3 dt 24) dx xln x6 4
Solve the problem. 25) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = -5t + 9, s(0) = 12 Find the area of the shaded region. 26) f(x) = -x3 + x2 + 16x g(x) = 4x 27) y = sec2 x y = cos x 5
Find the volume of the solid generated by revolving the shaded region about the given axis. 28) About the y-axis y = 5x Set up an integral for the length of the curve. 29) x = y2 + 8y, 0 y 8 Find the area of the surface generated by revolving the curve about the indicated axis. 30) 4x - x2, 0.5 x 1.5; x-axis 31) 4xex dx Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) xk, using the indicated point in the kth subinterval for ck. 32) f(x) = x2-2, [0, 8], left-hand endpoint k=1 6
Find the derivative. d x6 33) dx sin t dt 0 34) sin (9x - 10) dx Solve the problem. 35) The amount of alcohol in the bloodstream, A, declines at a rate proportional to the amount, that is, da dt = - ka. If k = 0.4 for a particular person, how long will it take for his alcohol concentration to decrease from 0.10% to 0.05%? Give your answer to the nearest tenth of an hour. Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 36) y = (x + 3)x Find the value of df-1/dx at the given value. 37) f(x) = 2x2, x 0, a = 16 38) 6 cos3 3x dx 7