Final Problem Set Name A. Show the steps for each of the following problems. 1. Show 1 1 1 y y L y y(1 ) L.. Use the information in #1 to show a solution to the differential equation dy y ky(1 ), where k and L are constants and e c L b, is y dt L 1 kt be. 3. Find the limit of y = f(t) as t. 3 4. Given y g( t), identify L, k and b. Find the initial condition. t 1 4e 5. Verify that the equation satisfies the differential equation in #. 6. Use your calculator to sketch a graph of y = g(t). 7. Calculate y" d y. Find the point of inflection and justify your answer.
dy y B. The graph of the slope field for the equation.05y(1 ) is shown below. dt 800 1. Graph the solution curve for the initial condition y(0) = 00.. Graph the solution curve for the initial condition y(0) = 100. 3. Graph the solution curve for the initial condition y(0) = 800. 4. As t increases without bound, ( t ), what is the limit of each of the 3 functions? 5. Discuss the rate of growth from (0, ) for the differential equation with each initial condition. INSERT GRAPH
C. A state game commission releases 30 elk into a game refuge. After 5 years, the elk population is 94. The commission believes that the environment can support no more dp p than 3000 elk. The growth rate of the elk population p is kp(1 ), dt 3000 30 p 3000 where t is the number of years. 1. Write a model for the elk population in terms of t. (Find L, k and b.). Draw the slope field for the differential equation and the solution that passes through the point (0, 30). Use increments of 5 from 0 t 50. 3. Use the model to estimate the elk population after 0 years. 4. Find the limit of the model as t. 5. After how many years is the elk population growing most rapidly? Solve algebraically and support your answer with a graph.
D. Use a graphing calculator to investigate the effects of the values of L, k and b on the 1 graph of y. Include examples graphs to support your conclusions. Consider 1t 1 1e positive, negative and fractional values. How do L, k and b fit into the standard transformation form of g(x) = a. f (b(x-h)) + k?
E. Integration by parts. d dv du 1. Given: [ uv] u v = uv +vu where u and v are differentiable functions of x. Prove: udv uv vdu Using the above rule, solve the following integrals. x. x e 3. x sin( x) 4. x ln( 3x) 1 1 5. sin ( x) 0 3 6. e x cos(x) 7. ln(1 x ) x x f ( x) e sin( x 8. Graph ) for x in [0, π]. a. Find the extrema and point(s) of inflection (algebraically). b. Find the area of the region bounded by f(x), y = 0, x = 0 and x = π. c. Find the volume of the solid generated by revolving f(x) about the x-axis.
F. Integrals Involving Powers of Sine and Cosine Find the following integrals. Show your work. 1. sin( x) sin( x ) cos( x) sin( x ) cos ( x). sin 3 (x) sin 3 3 4 ( x ) cos( x) sin ( x ) cos ( x) 3. sin 5 (x) sin 5 5 ( x ) cos( x) sin ( x ) cos ( x) Find the following integrals. Show your work. 4. cos( x) cos( x )sin( x) cos( x ) sin ( x) 5. cos 3 (x) cos 3 3 4 ( x ) sin( x) 6. cos 5 (x) cos 5 5 ( x ) sin( x) Use the trigonometric identities for sin (a + b), cos (a + b), sin(x) and cos(x) to find the following integrals. Show your work. 7. cos (x) sin (x) 4 8. 4 sin ( x ) cos ( x) 4 4
G. Everyday CALCULUS Discovering the Hidden Math All around Us by Oscar E. Fernandez 1. Find 3 headlines (include them here) in a newspaper, magazine or news source that are talking about rates of change and illustrate each graphically.. What everyday event makes you think mathematically? Explain. 3. Choose a chapter in the book and write a one page passage in the same style to describe your event (from #). Be sure to cite the chapter. 4. In the epilogue, the author gives a takeaway for each chapter. Write your own takeaway for the book.