Math 132 Information for Test 2

Transcription

1 Math 13 Information for Test Test will cover material from Sections 5.6, 5.7, 5.8, 6.1, 6., 6.3, 7.1, 7., and 7.3. The use of graphing calculators will not be allowed on the test. Some practice questions are given below. You may also wish to review the crucial differentiation and integration formulas available at jvanderw/classes/m13s11/calrulesii.pdf I. Inverse trigonometric functions and hyperbolic functions. The following are some practice problems over these basic topics. 1. Differentiate: (a) f(x) = e x arctan(x ) (b) g(x) = x arcsin x + 1 x. Find relative extrema of f(x) = arcsec x x. 3. A billboard 85 feet wide is perpendicular to a straight road and is 0 feet from the road. Find the point on the road at which the angle θ subtended by the billboard is a maximum.. Find the area under the curve 1 x x + 5 for x Evaluate the integral π 0 cos t 1 + sin t dt. 6. Evaluate 8x dx x x 7. Prove the identities (a) cosh x sinh x = 1 (b) sinh x = sinh x cosh x (c) sinh x = cosh x 1 8. Using their definitions, find the derivatives of cosh x, sinh x and tanh x. 9. Show sinh x is odd and increasing. Use its definition to find sinh 1 x. 10. Find the derivative of sinh 1 x using the definition you found in 9. 1

2 II. Differential Equations and Slope Fields. Key types questions: slope fields, solutions through a specific point, checking whether something is a solution, exponential growth, the logistic equation, Newton s law of cooling, radioactive decay, Carbon dating, separation of variables. The following are some basic practice problems. 1. Consider the slope fields given below. (i) (ii) (iii) (iv) Match the slope fields above with the appropriate differential equation. (a) dy dt = 1 y(y ) (b) dy dt = y t (c) dy dt = 1 (t 1)(t + ) (e) dy dt = (t + )(1 y) (g) dy dt = 1 y( y ) dy (d) dt = t y dy (f) dt = (t + ) (1 y) (h) dy dt = 1 (1 t)( + t)

3 . Consider y = xy y. (a) Where do you expect to see horizontal slopes? (b) Complete the table x y dy/dx (c) We used to generate the slope field for this differential equation given below. On the slope field, sketch the solutions through the points (i) (, 0) and (ii) (1, 3) 3. Solve the differential equation dy dx = 3yx yx subject to y(1) = 5. 3

4 . The solution to the logistic equation dy (1 dt = ky y ) can be written y = L L > 0 and k > 0. (a) Find lim t y(t). L where 1 + be kt (b) Use the method of separation of variables to verify the solution listed above is correct (this question will be postponed until the final, because in one step the integration uses partial fractions) (c) Write the solution to the logistic equation dy dt = y 5 y 150 that has initial condition (0, 8). 100 (d) A logistic equation has solution y(t) =. Find: (i) the value of k; (ii) the 1 + 9e 0.75t carrying capacity; (iii) the initial populaton; (iv) when the population will reach 50% of its carrying capacity; and (v) write the differential equation that has solution y(t) as given. 5. When an object is removed from a furnace and 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 110 F. Use Newton s law of cooling to find the core temperature 5 hours after the object is removed from the furnace. 6. The number of bacteria in a vat of potato salad is growing exponentially. After the potato salad had been sitting in the buffet for hours there were 15 bacteria present and after hours there were 350 bacteria present. Assuming no one takes any potato salad because it smells funny. Find: (a) the initial population of the bacteria; (b) the exponential growth model for the bacteria where t is measured in hours; (c) the number of bacteria the model predicts will be present after 8 hours when the buffet closes; (d) after how many hours the bacteria count will be 5, An archaeological dig finds a bone that has 68% of its original Carbon-1 remaining. Determine the age of the bone using a half-life of 5715 years for Carbon-1. If the bone is placed in a museum, what percentage of its original Carbon-1 will remain after another 1500 years? 8. Find the general solution to the differential equation dy dx = 5 6y (x + 1) 9. Which of the following are solutions to the differential equation y + y + 5y = 0? (a) y = e x (b) y = e x cos x (c) y = x

5 III. Applications of Integration 1. (a) Find the tangent line to the curve y = 1 x + 1 when x = 1. (b) Find the area of the region that is bounded by the curve and its tangent line in (a).. Find the area bounded by the line y = x + and the curve y = (x + ) Find the area between the curves x = y and x = y.. Revolve the line y = r x for 0 x h about the x axis to create a right circular cone on h its side with base of radius r and height h. Use the method of disks to find the volume of the cone. 5. Consider the region 0 y h h x for 0 x r. Describe the solid obtained by revolving r this region about the y-axis, and find its volume using the method of shells. 6. Two cylinders of radius r intersect at right angles (see the figure below). Use the method of slices to find the volume of the solid of intersection. 7. Use the method of your choice to find the volume of the solid formed when the region bounded by y = x and y = x is revolved about (a) the line y = 3; and (b) the line x = find the volume of the solid formed by revolving the region bounded by the graphs of y = x 3 + x + 1, y = 1 and x = 1 about the line x =. 9. A torus is formed by revolving the region bounded by the circle x + y = 9 about the line x = 5. Find the volume of the torus. 10. Use the method of cross-sections to find the volume of a pyramid with square base of area 5 units and height 3 units 5