Advanced Placement Physics C Summer Assignment

Transcription

1 Advanced Placement Physics C Summer Assignment Summer Assignment Checklist: 1. Book Problems. Selected problems from Fundamentals of Physics. (Due August 31 st ). Intro to Calculus Packet. (Attached) (Due August 31 st ) 3. Pre AP/AP 017 Summer Assignment, Part 1. (Due August 31 st ) 4. Pre AP/AP 017 Summer Assignment, Part. (Due September 5 th ) Course Syllabus for AP Physics C This content is treated at a college calculus based physics course level and students should be enrolled in AP Calculus also. TWO 1 ½ hour AP exams will be taken and students will be eligible for credit for TWO college classes. I. Classical Mechanics (Reviewed from AP Physics 1 with an emphasis on applications of calculus.) A. Kinematics B. Forces in Equilibrium C. Constant Force D. Energy E. Momentum F. Rotational Motion G. Oscillation H. Universal Gravity II. Electricity and Magnetism A. Electrostatics B. Current Electricity C. Electromagnetism

2 AP Physics C SUMMER ASSIGNMENT (BRING TO CLASS ON THE 1 ST DAY OF SCHOOL!): Part 1: Book problems: The following problems from your textbook Fundamentals of Physics should all be review from this year. Allow yourself plenty of time to complete them, as many have multiple portions. Show all steps clearly on paper, as the first online assignment in August will relate to this assignment. Show clear work leading to the answer. Please come prepared in August to discuss any problems that gave you a great amount of difficulty, or that you were unable to solve. (Answers to odds are in back of the textbook, answers to evens are attached to the end of this packet. Please note that the problems are all from the sections marked Exercises and Problems Chapter pg : 4, 69, 73, 90 Chapter 3 pg. 54: 3, 5 Chapter 4 pg. 80: 35, 37 Chapter 5 pg : 17, 7, 34, 53 Chapter 6 pg : 3, 9 Chapter 7 pg : 15, 0, 37, 67 Chapter 8 pg : 3, 10 Chapter 9 pg : 35, 4 Part : Intro to Calculus Packet: These are the things you need to know how to do. You will learn the theory in your calculus class, but a little bit earlier can t hurt either! Part 3: Complete Part 1 of the Pre AP and AP 017 Summer Assignment packet (Due August 31 st ) and research and write your Research Plan for next year s research project in your Project Journal. (Due September 5 th ) Write all information in your Project Journal under the Choosing a Project tab. Resources for this assignment are linked off the school homepage. Good luck and have a great summer! Feel free to contact me by r.sweeney@schoolsofwestfield.org or through Remind if you have questions or would like me to review your Research Plans in advance. Introduction to Calculus x t Let s start by looking at a graph of position as a function of time (x vs. t). x t1 Slope = t 1 t y x xt xt x t t t 1 1 v t t We know that we can find the average velocity between two points (t 1 and t ) by finding the slope of the line connecting those two points.

3 This approach, however, does not give us a very accurate representation of the instantaneous velocity at every point in between t 1 and t. In order to find the instantaneous velocity at any point on our line (t 3), we must make the time interval very small. There are several approaches we can take to accomplish this. First, we can find the slope of the tangent line at t 3. A second approach could be to take the limit as t 0. t 3 t slope x x( t t) x( t) lim lim t t t0 t0 xδt xt xδt xt xδt xt Δt t Δt t The trick or short cut to solving derivatives is; Δt t The procedure described above is known as a derivative. n if x() t at, where a is a constant and n is any positive or negative number (integer or fraction), then the n1 derivative of x with respect to t is nax. dt Rules: If x(t) is a polynomial or algebraic function of x, we apply this expression to each term in the polynomial. Ex. if Ex. if 3 d( at ) d( bt), then 3at b dt dt dt x( t) 3t 10t, then d 3t 10t d 3t d 10t 6t 10 dt dt dt dt 3 x() t at bt If x is a constant, then 0 dt. Ex. if xt ( ) 4, then 0 dt.

4 Power Rule Derivatives: (For each of the following functions, find the derivative (dy/) with respect to x. 1. a. y= 6x 4 b. y = 7x c. y = x 9 d. y = 7 e. y = 5x -1 f. y = 5x 3 + 8x

5 . For each of the following functions, find the derivative (dy/) with respect to x: a. y = cos x b. y = e x c. y = ln x 3. Use the chain rule to help you find dy/ for each of the following functions. (9 Points) a. y = (x + 4) 3 b. y = e x c. y = sin (x + 5)

6 4. Use the product rule to help you find dy/ for each of the following functions. (9 Points) a. Y = (x )(x - 7) b. Y = x 3 e x c. Y = (5x + 3x)(ln x) 5. Find the maximum y-coordinate reached by the following functions: a. y = -3x + 1x b. y = -x -0x + 1

7 6. John is traveling to work in traffic one morning and is very bored. He decides to take measurements along his route to pass the time. John finds out that his motion can be explained by the relationship, x t t t. 1 3 ( ) 3 0 John needs your help in determining how fast he was traveling 3 seconds into his trip. a. Using a full sheet of graph paper, carefully draw a graph of b. Draw a best fit curve of your graph. x t t t. 1 3 ( ) 3 0 c. Using your graph, find John s average velocity for the first six seconds of the trip. Do this by connecting the zero second mark and the six second mark on your graph with a straight line and calculating the slope of that line. The slope will be the average velocity. (Record this on the graph) d. Repeat this again for the one to five second interval and two to four second interval. (Record this on the graph) e. Now find John s instantaneous velocity at three seconds by drawing a line tangent to the curve at that mark and finding the slope of that line. (Record this on the graph) f. Using the Power Rule, find the derivative of the equation mapping John s trip. dt g. Plug 3 seconds in for t and record your answer. Instantaneous slope of function at 3 seconds = h. Compare this answer to the instantaneous velocity found graphically and the three average velocities.

8 Introduction to Integrals While differential calculus stems from the problem of finding the slope of a curved line, integral calculus comes from how to calculate the area under the f(x) curve. Starting with a graph of f(x) = 3x, the area of the curve of f(x) = 3x between points a and x would be A(x) = 3(x-a) But what happens if the graph is less obvious? For example, what if f(x) were f(x) = 4x? The idea is that you look at many many little rectangles, and if you add these up they will approximate the area under the curve. As the bottom length of the rectangles (Δx) goes to zero, the sum of the areas of the rectangles becomes closer to the actual value. So we want to take the integral of the function f(x) in order to sum all the little areas ΔA to come up with the total sum An integral is basically the opposite of a derivative and is expressed by the symbol If you want to take the definite integral (for example to find the area under the curve f(x) = 3 from 1 to 5, the integral would be written Some rules for computing integrals: 1. The integral of a constant c with respect to x would be written c and is equal to cx For example: 5 5x

9 For example: For example:. The integral of x n would be written x n and is equal to 6 5 x x 6 1 x n n 1 3. The integral of a f(x), where a is a constant, is equal to a times the integral of f(x). (Basically you can factor out the constant) 3x 3 x x 3 3x 4. The integral of f(x) ± g(x) = f ( x) g( x) 7. Use the idea of un-doing a derivative to find the following indefinite integrals: a. 6 x b. (1x 3 x) ) 3 c. (6x 1 8. Evaluate each of the following definite integrals: (8 a. x 3) 0 5 (7 b. x 1) 1

10 9. Use calculus ideas to find the required graphical values of the following functions: a. Find the instantaneous slope of the function y 3x 6x 9 at the point (,5) b. For the same function ( y 3x 6x 9 ), find the area under the curve between x = 0 and x =. SUMMER EXTRA CREDIT: Have a picture taken of you in front of a nationally famous landmark or with a famous person. You must be holding a sign with a physics equation on it, and the sign must be legible in the photo. Bring the photo to school during the first week.