Calculus II/III Summer Packet

Transcription

1 Calculus II/III Summer Packet First of all, have a great summer! Enjoy your time away from school. Come back fired up and ready to learn. I know that I will be ready to have a great year of calculus with all of you. These problems represent a review of topics from second semester AP Calculus BC that will not be covered in detail during the first semester of Calculus II. You will need to be familiar with all of these topics. In addition to the topics represented in this packet, you should be very familiar with the unit circle and with the derivatives and integrals of elementary functions including trig functions and their inverses. Calculus II/III will not emphasize the use of calculators. We will do the vast majority of problems during the course without a calculator. I know that your calculator can do some calculus. I want you to be able to do calculus. Do the problems in this packet at your leisure, but do the problems! They are beneficial. We will not go over this packet in class. Solutions will be available after the start of the semester. My assumption will be that everyone has completed the packet and is familiar with the material. If you have questions about specific problems from the packet, me, ask in class or see me individually outside of class at the beginning of the year. Finally, I love math. All kinds of math. If you have questions about or interests in topics that we may not cover in class, talk to me. I will find a way to incorporate them into the year.

2 Area between curves. Volume by disks, washers and shells Multiple solution methods may work for the volume problems. 1. Find the area bounded by the parabola y = x 2 3x and the line y = x. 2. Find the area between the curves x = 2(y 1) 2 and (y 1) 2 = x Find the volume of the solid generated by revolving the region bounded by y 2 = 4x and the x axis from x = 0 to x = 4 about the x axis. 4. Find the volume formed by revolving the hyperbola xy = 6 from x = 2 to x = 4 about the x axis. 5. Find the volume of the solid of revolution formed by revolving about the y axis the region bounded by y = x 2 for x values on [0, 2]. 6. Find the volume of the solid generated by revolving the region between y = x 2 + 6x 8 and the x axis about the y axis.

3 Exponential and logarithmic functions 1. Find the derivative. y = x 2 e x y = ln (1 + e 2x ) g(t) = t 2 2 t h(θ) = 2 θ cos πθ f(x) = log 2 x 2 x 1 f(t) = t 3 2log 2 t Find the integral. e x x 5 e dx 1 e x e 2x dx 3 x dx 32x dx x 5 x dx 2 sinx cosxdx 3. Use logarithmic differentiation to find dy dx. y = x 2 x y = (x 2) x+1 4. Solve the separable differential equation. y = x(1 y) y = 5x y

4 5. Use the given information to write and/or solve the differential equation that models the situation. All constants can be determined using the information in the problem. a. In radioactive decay, the rate of decay is proportional to the amount of radioactive material. Radioactive radium has a half-life of ~ 1599 years. What percent of a given amount of radium remains after 100 years? b. When a population in a given environment cannot grow past an upper limit L, called the carrying capacity, the differential equation that models the growth of the population is the logistic equation: dy dt = ky 1 y, k, L > 0 L Solve the logistic equation above given the following conditions to find the population y as a function of time: A conservation organization releases 25 panthers into a game preserve with a carrying capacity of 200 panthers. After 2 years, there are 39 panthers. Hint: the equation is separable and requires partial fractions. The solution requires some work! Do the work.

5 Integration Techniques You can check your work with a calculator, but you need to be able to do these by hand. 1. Use integration by parts to find the integrals. a. xe x dx b. x 2 lnx dx c. x 2 sinx dx d. sin 3 x dx 2. Use trigonometric techniques to find the integrals. a. sin 2 x cos 3 xdx b. sin 3 3x cos 5 3xdx. c. tan 3 3x sec 4 3xdx d. tan 4 x dx dx 3. Use trig substitution to find the integral. x 2 9 x 2

6 4. Use partial fractions to find the integral. If you use the cover up method, indicate that in your solution. The cover up method won t work for some of these. a. dx x 2 9 b. xdx x 2 3x 4 c. 6x+7 (x+2) 2 dx d. x2 +3x 4 x 2 2x 8 dx 5. Evaluate the improper integral. 4 a. dx 0 4 x 0 b. e 2x 4 c. dx 0 3 x 1