Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental Theorem of Calculus I) (Fundamental Theorem of Calculus II) d [ln u ] = dx 1 u du = tan u du = cot u du = sec u du = csc u du = Suppose f g are inverse functions the domain of f is an interval. If f is differentiable, then g (x) = d dx eu = e u du = For a > 0, a 1, a x = e?, d dx au = a u du = For a > 0, a 1, log a x = d dx [log a u] =
1. Evaluate π/3 0 [2 cos 2 x + 2 sin 2 x 3 tan x sec x 6x 2 ] dx. 2. Find the area of the region that lies below y = x 2 + 20x 64 above the x-axis. 3. (a) Suppose f (x) = (x + 2) x 1, find f. (b) Evaluate 5 2 (x + 2) x 1 dx 4. Use the Fundamental Theorem of Calculus to find the derivative of h(x) = sin(x) 2 (cos(t 4 ) + t)dt 5. Find an equation for the function f(x) that has the given derivative initial value. f (x) = x(x 2 + 1) 5 f( 1) = 13 3 6. (a) Use the trapezoidal rule with n = 6 to estimate π 0 sin x dx. (b) Use the trapezoidal error formula to determine the largest possible error you would expect in your answer to (a) (you will be given the error formulas for Simpson s Rule the Trapezoidal Rule if needed on the final test)? 7. Let y = (ln x) cos x for x > 1. Find dy dx then find the tangent line at (e, 1). 8. Let f(x) = x 2 ln(10 2x 2 ). Find f (x) use interval notation to give the domain of f.
9. (a) State the definition of ln x as given in the text. (b) Show that ln(ab) = ln a + ln b when a > 0 b > 0. (c) Use the definition of ln x to show for integers n 2 (draw a sketch!) 1 + 1 2 +... + 1 n 1 ln n 1 2 + 1 3 +... + 1 n (d) Verify that ln x is strictly increasing concave down for x > 0. 10. Differentiate f(x) = x7 3x2 2x+1 g(x) = log 7 [(x 2 + 7)(4x 4 + x 2 + 3)] 11. Find the average value of an integral function f(x) = 7 ln x x on the interval [1.1, 1.8]. 8x 3 + 2x 2 + 50x + 12 12. Evaluate dx x 2 + 6 13. Find the slope of the tangent line to the curve at the point (7π/2, 5π) cos(3x 3y) xe x = 7π 2 e 7π/2 14. (a) Let f(x) = 3x + 6x 11. Find a value a such that f(a) = 9, then find (f 1 ) ( 9). (b) Let f(x) = x 2 17x + 76 on the interval [8.5, ). Find a value a such that f(a) = 10, then find (f 1 ) (10). 15. Evaluate the integral (4 x) 5 5 (4 x)6 dx e x 1 16. Evaluate x 2 dx 17. Let f(x) = 5ex 7 19e x + 11. Find f 1 (x) find the domain of f 1. 18. Let f(x) = 8 x log 3 x, find f (x).
II. Topics Covered on Test II 1. Practice on the new differentiation/integration formulas: (a) Differentiate the following with respect to x: (i) arcsin u (ii) arctan u (iii) arcsec u (b) State the integral forms of the differentiation rules in (a) (c) State the definitions of sinh x, cosh x tanh x (d) Differentiate sinh x, cosh x tanh x. Then write the integral form of these differentiation rules. (e) Applications of some of the differentiation rules you stated above: Differentiate: arctan(e x + sin x) e 2x cosh(5 x 3 ) 5x 2 arcsin(e 4x ) (f) Applications of some of the integration rules you stated above: e x Evaluate: cosh(5x) dx dx x sinh(x 2 ) dx 1 + e2x (g) Use the definitions of sinh x cosh x given above to show (i) cosh 2 x sinh 2 x = 1 (ii) sinh 2x = 2 sinh x cosh x cosh(3x) dx 1 + sinh(3x) 2. A rectangular billboard 7 feet in height sts in a field so that its bottom is 8 feet above the ground. A cow with eye level at 4 feet above the ground sts x feet from the billboard, as illustrated in the diagram below. (a) Express θ, the vertical angle subtended by the billboard at her eye, in terms of x. (b) Find the distance the cow must st from the billboard to maximize θ.
sinh 7x 3. Evaluate the integral dx 7x 4. Evaluate 8x dx 2x2 x 4 5. An unknown radioactive element decays into non-radioactive substances. In 660 days the radioactivity of a sample decreases by 68 percent. (a) What is the half-life of the element? (b) How long will it take for a sample of 100 mg to decay to 90 mg? 6. Suppose that news spreads through a city of fixed size 200000 people at a rate proportional to the number of people who have not heard the news. (a) Formulate a differential equation initial condition for y(t), the number of people who have heard the news t days after it has happened. ( b) 5 days after a scal in City Hall was reported, a poll showed that 100000 people have heard the news. Using this information the differential equation, solve for the number of people who have heard the news after t days. 7. When an object is removed from a furnace placed in an environment with a constant temperature of 80 F, its core temperature is 1500 F. One hour after it is removed, the core temperature is 1120 F. Use Newton s law of cooling to find the core temperature 5 hours after the object is removed from the furnace. 8. Solve the differential equation dy dx = (x 4)e 2y then find the particular solution that satisfies y(4) = ln(4). 9. Solve the differential equation dy dx = 88xy10 subject to y(0) = 4. 10. Which of the following differential equations generates the slope field given below? (a) dy dx = xy y dy (b) dx = y xy dy (c) dx = x xy (d) dy dx = xy x
11. Biologists stocked a lake with 233 fish estimated the carrying capacity to be 9700. The number of fish tripled in the first year. Assuming the fish population satisfies the logistic L equation P (t) =, find P (t) determine how long it will take for the population 1 + be kt to reach 4850. 12. Find the area between the curves x = 4 y 2 x = y 2. 13. Find the area bounded by the line y = x + 2 the curve y = (x + 2) 3. 14. The region between the graphs of y = x 2 y = 2x is revolved around the line y = 4. Find the volume of the resulting solid. 15. A ball of radius 17 has a round hole of radius 3 drilled through its center. Find the volume of the resulting solid. 16. (a) Use the method of volume by cross-sectional slices to find the volume of a pyramid with a square base of 260 cubits a height of 1000 cubits. (b) Repeat problem (a) where the base has side b height is h b h are unknown constants. 17. Find the volume of the solid obtained by revolving the region bounded by the curves y = 1/x 3, y = 0, x = 3 x = 5 about the line y = 3. 18. Use the method of your choice to find the volume of the solid formed when the region bounded by y = x 2 y = 2x is revolved about (a) the line y = 3; (b) the line x = 7. 19. Differentiate: (a) f(x) = e x arctan(x 2 ) (b) g(x) = x arcsin x + 1 x 2 20. Suppose arcsin(x 3 y) = xy 3. Use implicit differentiation to find dy dx. 21. Evaluate the expression tan(2 cos 1 (x/3)). 22. To solve the equation (cos x) 2 1.1 cos x 1.26 = 0.
III. Topics Covered on Test III 1. Find the arc length of the curve y = 1 10 ( e 5x + e 5x) for 0 x 3 2. Find the area of the surface obtained by rotating the curve y = 9x 3 for x = 0 to x = 10 about the x-axis. 3. Set-up evaluate an integral to find: (a) The arc length of the curve y = x3 6 + 1 2x, 1 x 2. (b) The surface area obtained by revolving the curve in (a) about the x-axis. 4. A tank containing water has a trapezoidal cross section. The width of the edge of the top of the trapezoid is b 2 = 2 meters, the width of the base of the trapezoid is b 1 = 1.5 meters the height of the trapezoid is h = 1.8 meters. The depth of the water in the trough is d = 1.4 meters (see the diagram below that is not drawn to scale), the length of the trough is 5 meters. Find the work required to pump the water out over the top edge of the trough. Use the fact that the water weighs 1000 kg per cubic meter. 5. Consider the planar lamina bounded by y = x 2 +4x+2 y = x+2 with uniform density ρ. Find M, M x, M y ( x, ȳ). Set-up the integrals then use technology to evaluate the integrals. 6. The cross-section of the end of a trough is described by x 4 y 1 where 1 x 1. Then the width of the top of the cross-section of the trough is 2 feet, the length of the trough is 10 feet. The trough is full of pig swill that weights 50 pounds per cubic foot. (The cross-section of the end of the trough is shown in the figure below). (a) Find the fluid force of the swill on one end of the trough.
(b) Find the amount of work required to pump all of the swill over the top edge of the trough. 18x 3 + 5x 2 + 128x + 35 7. Evaluate the integral dx. x 2 + 7 8. Use integration by parts to evaluate ln x dx x n ln x dx where n 1. happens when n = 1? 9. Use tabular integration by parts to evaluate x 3 e 5x dx. What 10. Evaluate the trigonometric integral 11. Evaluate the integral π/5 1 x 2 121 x 2 dx. 0 sin(2x) sin x dx. 12. For each of the following integrals, find an appropriate trigonometric substitution of the form x = f(t) to simplify the integral. (a) (8x 2 4) 3/2 dx. (b) (c) (d) x 2 5x2 + 2 dx. x 7x 2 + 42x + 60 dx. x 50 6x2 36x dx. 13. The velocity of a particle moving along a line is given by v(t) = t sin 2 (4t) for 0 t 3. Find the distance travelled by the particle. 14. Evaluate the integral sin 3 (18x) cos 8 (18x) dx. 15. Evaluate the integral 16. Evaluate the integral tan 3 (9x) sec(9x) dx. 105x 6 sec 4 (x 7 ) dx. 17. Find the arc length for the curve y = ln(4x) where 1 x 9. 18. Evaluate the integral 41 45 96x 64x 2 dx.
IV. Questions from Section 8.5 to 8.8 1. Evaluate the integral 3 2 x 3 2 (x + 8)(x + 7) dx. 9x 3 + 7x 2 + 100x + 125 2. Evaluate the integral dx. x 4 + 25x 2 5 3. Use the table of integrals in your text to evaluate 4x 4x2 dx 4. Use the table of integrals in your text to evaluate 3 1 dt 9t2 1. 5. Evaluate the limit lim x 0 e 8x 1 sin(13x) ( 6. Evaluate the limit lim 1 + 6 ) x 15. x x ( 5 7. Use L Hopital s rule to evaluate lim x 1 + ln x 5 ). x 1 8. Determine whether the integral determine its value. 13 3 13 3 x 3 dx converges or diverges. If it converges 9. Determine whether the integral xe 3x dx converges or diverges. If it converges, determine its value. 3 10. Determine whether the integral dx converges or diverges. If it converges, determine its value. 4 ln x x 11. Determine whether the improper integral determine its value. 3 3 1 dx converges or diverges. If it converges, x4