Review: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 Review: A Cross Section of the Midterm Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it eists. 4 + ) lim - - ) A) - B) - C) 4 D) ) lim ) A) 9 8 B) C) 4 D) Determine the limit algebraically, if it eists. - 4 ) lim - + A) Does not eist B) -4 C) - D) ) Use l'hopital's Rule to evaluate the limit. cos 9-4) lim A) 9 B) 8 C) - 8 D) 4) Find a value for a so that the function f() is continuous. ) f() = -, < 4 a, 4 A) a = 4 B) a = 9 C) a = D) a = ) Find dy/d by implicit differentiation. If applicable, epress the result in terms of and y. ) cos y + = y A) 4 - sin y y4 B) 4 + y sin y y4 - sin y C) 4 - y sin y y4 + sin y D) 4 + sin y y4 ) Solve the problem. 7) The position of a particle moving along a coordinate line is s = + t, with s in meters and t in seconds. Find the particle's velocity at t = sec. A) - m/sec B) m/sec C) m/sec D) m/sec 7)

2 Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f()g()f ()g () 8) 8 8) f() + g() at = A) 7 B) 7 C) - 7 D) 7 At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 9) y - π cos y = π, slope at (, π) A) B) -π C) π D) - π 9) Find the derivative of the given function. ) y = sin- (4) A) - 8 B) - 8 C) - 8 D) - 4 ) Find the value of df-/d at = f(a). ) f() =,, a = A) B) 4 C) D) ) Find dy/d. ) y = ln (ln ) A) B) ln C) D) ln )

3 Find the derivative at each critical point and determine the local etreme values. ) y = /( - ); A) Critical Pt. Derivative Etremum Value = = maimum minimum ) B) Critical Pt. Derivative Etremum Value = Undefined local ma 4 = minimum C) Critical Pt. Derivative Etremum Value = Undefined local ma = minimum D) Critical Pt. Derivative Etremum Value = Undefined local ma = minimum.748 Use the Concavity Test to find the intervals where the graph of the function is concave up. 4) y = 4 4) A) (, ), (-, ) B) (-, ) C) (, ) D) (-, ) Find the points of inflection. ) y = ) A) (, 4) B) (, ) and (, 4) C) (, ) and (,-8) D) (, )

4 Solve the problem. ) Select an appropriate graph of a twice-differentiable function y = f() that passes through the points (-,), -, 9,(,),, and (,), and whose first two derivatives have the 9 following sign patterns. ) y : y : A) B) y y C) D) 4 y y

5 7) A rectangular sheet of perimeter cm and dimensions cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of and y give the largest volume? 7) A) = cm; y = 9 C) = cm; y = 7 cm B) = cm; y = cm cm D) = cm; y = cm 8) A container is the shape of an inverted right circular cone has a radius of. inches at the top and a height of. inches. At the instant when the water in the container is. inches deep, the surface level is falling at the rate of -.9 in./s. Find the rate at which water is being drained. A) -. in./s B) -. in.s C) -.7 in./s D) -.8 in./s 9) An engine cylinder. cm deep is being bored such that the radius increases by.4 mm/min. How fast is the volume V of the cylinder changing when the diameter is 7. cm? A). cm/min B) 84 cm/min C).4 cm/min D) 4 cm/min ) A man ft tall walks at a rate of 7 ft/s away from a lamppost that is 8 ft high. At what rate is the length of his shadow changing when he is 7 ft away from the lamppost? A) 7 4 ft/s B) ft/s C) 7 4 ft/s D) 7 8 ft/s 8) 9) ) Graph the integrand and use areas to evaluate the integral. ) ( - ) d - A) B) C) 7 D) ) Solve the problem. 7 7 ) Suppose that f is continuous and that f(z) dz = and f(z) dz =. Find - f() d. - - A) - B) - C) - D) )

6 Evaluate the integral. ) 4 - d /4 A) - ln.7 B) - ln C) - ln. D) - ln ) Find dy/d. 4) t dt 4) A) / B) C) 4 - D) 84 Find the average value over the given interval. ) y = 7 sin ; [, π] A) B) 4 π π C) 98 π D) 7 π ) Find the total area of the region between the curve and the -ais. ) y = - + 9; 4 ) A) 4 B) 7 C) D) Solve the problem. 7) Suppose that f is the differentiable function shown in the graph and that the position at time t t (in seconds) of a particle moving along a coordinate ais is s = f() d feet. y = f() y 8 (8, 8) 4 (,.4) (, ) (4, -.4) -4 (, -.) ) At what time during the first 7 sec does s have its smallest value? A) sec B) 7 sec C) sec D) sec

7 8) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use the Trapezoidal Rule to approimate the distance traveled by the car in the 8 seconds. 8) Time (sec) Velocity (ft/sec) A) 4. feet B). feet C) 9 feet D) feet 9) Suppose that f(t) dt = Find f(). A) + 7 B) ) C) D) ) Find the linearization of f() = 9 + tan πt 4 dt at =. A) 9 + B) + 9 C) 9 D) + 9 ) Use Euler's method to solve the initial value problem. ) dy = y - and y = 4 when = d Use Euler's method with increments of Δ =. to approimate the value of y when =.. A).49 B).749 C).799 D).9 ) Solve the initial value problem eplicitly. ) dy d = e - cos and y = when = A) y = e - sin + B) y = e - sin - C) y = e + sin - D) y = e+ - sin + ) 7

8 Solve the problem. ) A wild animal preserve can support no more than elephants. elephants were known to be in the preserve in 98. Assume that the rate of growth of the population is dp =.7P( - P) dt where t is time in years. Find a formula for the elephant population in terms of t. A) P = where t is the number of years since e-.84t B) P = C) P = D) P = +.4e-.84t -.4e-.84t +.4e-.84t where t is the number of years since 98 where t is the number of years since 98 where t is the number of years since 98 ) Evaluate the integral using the given substitution. 4) cos () d, u = 4) A) sin ( ) + C B) sin (u) + C u C) sin() + C D) sin ( ) + C Evaluate the definite integral by making a u-substitution and integrating from u(a) to u(b). e d ) 4 + e A) e ln 4 + e B) ln 4 + e C) ln 4 + e D) e ln(4 + e) 4 ) Evaluate the integral. ) sin d A) cos - sin + C B) - cos + sin + C C) - cos - sin + C D) - cos + sin + C ) 7) y ln 7y dy A) y ln 7y - y + C B) 7 y ln 7y - ln 7y + C 49 C) y ln 7y - y 4 + C D) y ln 7y + y 7 + C 7) 8

9 Evaluate the integral by using a substitution prior to integration by parts. 8) + d A) -.94 B).9 C) -. D) -.7 8) The function v(t) is the velocity in m/sec of a particle moving along the -ais. Find the total distance traveled by the particle. 9) v(t) = 8 cos t, t π 9) A) B) 8 C) D) Solve the problem. 4) Population density measures the number of people per square mile inhabiting a given living area. The population density of a certain city decreases as you move away from the city center and can be approimated by the function 8(. - r) at a distance r miles from the city center. The radius of the city is. miles. Set up an integral that can be used to estimate the total population of the city.. π A) 8(. - r) dr B) 8(. - r)(πr) dr.. C) 8(. - r)(πr) dr D) 8(. - r)(πr) dr 4) Find the area enclosed by the given curves. 4) Find the area of the region in the first quadrant bounded by the line y = 8, the line =, the curve y =, and the -ais. 4) A) 4 B) C) D) 4 Find the area of the regions enclosed by the lines and curves. 4) - = y, = y + A) 49 4 B) 4 C) 8 D) 9 4) Find the volume of the described solid. 4) The base of the solid is the disk + y. The cross sections by planes perpendicular to the y-ais between y = - 4 and y = 4 are isosceles right triangles with one leg in the disk. A) 8 B) 4 C) D) 4 4) Find the volume of the solid generated by revolving the region about the given line. 44) The region bounded above by the line y = 9, below by the curve y = 9 -, and on the right by the line =, about the line y = 9 A) 4 π B) 48 7 π C) π D) 9π 44) 9

10 Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated ais. 4) About the -ais 4) 4 y y = = = 9 - y 4 A) 9π B) 7π C) 9 π D) 8π Set up an integral for the length of the curve. 4) y = -, A) ( + 4) d B) C) + (- )d D) + 4d + d 4) Find the eact length of the curve analytically by antidifferentiation. 47) y = + from = to = 47) A) 9 B) 4 C) 8 D) 7 Evaluate the improper integral or state that it diverges. d 48) 4 48) A) B) 4 C) D) diverges 49) dt t - t A) ln B) ln C) - ln D) ln 49)

11 Find the area or volume. ) Find the area of the region in the first quadrant between the curve y = e- and the -ais. ) A) e B) C) D) ) Find the volume of the solid generated by revolving the region under the curve y = 9, from ) = to =, about the -ais. A) π B) 9π C) 8π D) 9 9 ) Find the volume of the solid generated by revolving the region under the curve y = 4e- in the first quadrant about the y-ais. A) 4π B) π C) π D) 4π ) Find dy/d in terms of t. ) = t + t, y = t - t + A) t + t - B) t - t + C) t + t - D) t - t + ) Find an equation for the line tangent to the curve at the point defined by the given value of t. 4) = t, y = t, t = 8 A) y = 4 + B) y = - 4 C) y = 4 D) y = - 4-4) Find the length of the curve. ) = 7 sin t, y = 7cos t, t π A) 7 B) 4π C) 49π D) 7π )

12 Answer Key Testname: UNTITLED ) D ) C ) B 4) C ) C ) C 7) D 8) B 9) B ) B ) D ) D ) C 4) C ) B ) D 7) D 8) D 9) C ) A ) B ) B ) B 4) B ) B ) D 7) B 8) A 9) A ) D ) D ) B ) B 4) A ) B ) D 7) C 8) B 9) A 4) C 4) D 4) B 4) C 44) A 4) A 4) B 47) B 48) C

13 Answer Key Testname: UNTITLED 49) A ) D ) B ) D ) B 4) A ) D

Find the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis

Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais