Learning Communities in Mathematics (LiCMATH)

Transcription

1 Learning Communities in Mathematics (LiCMATH) The Pomona College Mathematics Department has established Learning Communities in Mathematics (LiCMATH, pronounced Like Math ) to provide an honors-level experience to well-prepared students in Calculus I and II. Every year we invite a diverse group of incoming students to apply to the program. Members of the Learning Communities in Mathematics are required to attend weekly LiCMATH workshops, during which they work on challenging Calculus problems with the help of each other, student mentors and a supervising faculty member. Additional mathematical and social events are held throughout the semester. Members of LiCMATH will also attend the usual mentor sessions for their classes. An adaptation of Uri Treisman's work at UC-Berkeley and the University of Texas, our Mathematics Learning Communities aim to provide a fun, challenging experience to students who are interested in and prepared for Calculus. We believe that LiCMATH students, like their counterparts at other institutions, will excel in Calculus and enjoy it. Alumni of similar programs at other schools report building some of their closest friendships in the program and go on to major or minor in mathematics, or a mathematically based field. While participation in LiCMATH is restricted, any interested student planning to take Calculus I or II (Math 30 or 31) is eligible to apply. Please fill out the form below.

2 Pomona College Learning Community in Mathematics (LiCMATH) Application Please your responses to the following items to the Professor Radunskaya by September 1, Your name: Preferred Your high school (including city): List the mathematics courses you have taken in high school or at a community college and the grades you received (list in reverse order, going back two years): AP Math or Stats Exams taken, and scores: What mathematics and science courses are you enrolled in this fall? What majors are you interested in? Your experience in mathematics classes: Describe (in a sentence or two) your experiences with mathematics and your attitudes toward the subject: Do you tend to study alone or in groups? In your math classes in high school, was there an emphasis on (check a box): Writing Word Problems Theory Group Work A lot Some None Don t know

3 Online Resources for Learning Communities in Mathematics Overview of Treisman s model: The Emerging Scholars Program Workshop s Calculus Problems Database: bin/prob.pl?math_problems_database Institutions with similar programs o UC Berkeley s Professional Development Program: o UT Austin s Emerging Scholars Program: scholars o UIUC s Merit Immersion for Students and Teachers (MIST): o St Mary s College of Maryland s Emerging Scholars Program:

4 Mathematics Learning Community - Calculus I - Fall, 2011 Worksheet #2 1. Let f(x) = 2x 1. (a) Express f (1) as a limit. (b) Find f (1) by evaluating this limit. 2. Let g(x) = x 3 1. Evaluate g ( 1), g (0), g (1). Hint: Recall that { y y 0 y = y y < 0. Sketch a graph of g(x) first. 3. Find a function f(x) whose derivative at any point a is given by Be very careful. lim [(3a + h 0 h)16/5 (3a) 16/5 ]/4h 4. (a) Imagine a road on which the speed limit is specified at every single point. In other words, there is a certain function L such that the speed limit x miles from the beginning of the road is L(x). Two cars A and B, are driving along this road; car A s position at time t is a(t), and car B s is b(t). (b) What equation expresses the fact that the car A always travels at the speed limit? (Hint: the answer is not a (t) = L(t).) (c) Suppose that A always goes at the speed limit, and that B s position at time t is A s position at time t 1. Show that B is also going at the speed limit at all times. (d) Suppose B always stays at a constant distance behind A. Under what circumstances will B still always travel at the speed limit?

5 Mathematics Learning Community - Calculus I - Fall, 2011 Worksheet #5 1. Assume that f(x) is a differentiable function and that the values of f(x) and its derivative at the points x = 0, 1, 2, and 3 are given by: f(0) = 3 f (0) = 1 f(1) = 5 f (1) = 0 f(2) = 2 f (2) = 3 f(3) = 6 f (3) = 1. Let g(x) = x 2 3x+2. For each function below, calculate the derivative at the given point. (a) f(x) + g(x); x = 0 (b) f(x) g(x) ; x = 1 (c) f(x)g(x); x = 2 f(x)g(x) (d) f(x)+g(x) x = 3 (e) f(g(x)); x = 0 (f) f(g(x)); x = 1 (g) g(f(x)); x = 2 (h) g(f(x)); x = 3 2. (Differentiating Inverse Functions ) (a) Prove that the formula for the derivative for an inverse function is (f 1 ) 1 (x) = f (f 1 (x)). (Hint: Let g(x) = f 1 (x), then f(g(x)) = x. Differentiate.) (b) Find f 1 (x) given that f(x) = 2x 3 x + 2. (c) Differentiate f 1 (x) from part b) and compare with the derivative you get by applying the formula in part a). 3. The speed limit on a stretch of highway is 55 mph. Highway patrol officer, Sgt. Corey, stations himself at a point, out of view of the motorists, 50 feet off the highway. Sgt. Corey is equipped with a radar gun which measures the speed at which a car approaches his position. He takes a reading of suspected speeders by pointing his radar gun at a point on the highway 120 feet from the point on the highway closest to him. The radar gun picks up a reading of 48 feet/sec for a green Chevy driven by Katie. How fast is she traveling? Is Katie speeding?

6 4. Javier and Luis have made themselves two dimensional! Javier moves along the positive horizontal axis, and Luis along the graph of f(x) = 3x, x 0. At a certain time, Javier is at the point (5, 0) and moving with speed 3 units/sec; and Luis is at a distance of 3 units from the origin moving with speed 4 units/sec. At what rate is the distance between Javier and Luis changing? Hint: Draw a graph!! 5. Find the derivatives of each of the following. Simplify your answer as much as possible. (a) y = sin(e x2 +x 1 ) (b) y = a x x a, a > 0 a, constant (c) y = 1 + e x (d) y = 2 (3x).

7 LiCMATH Math 31 Calculus II Worksheet 1 1. Consider f(x) = 2x x 2 on the interval [0, 2]. (a) Sketch the graph of f(x). (b) Partition the interval [0, 2] into subintervals of length 1. Using this partition: 2 i) Find a plausible way to compute an overestimate of the area bounded by f(x), the x axis, and the lines x = 0 and x = 2. ii) Find a plausible way to compute an underestimate of the area bounded by f(x), the x axis, and the lines x = 0 and x = 2. (c) Using the procedure you suggest above compute an overestimate and an underestimate of the area bounded by f(x), the x axis, and the lines x = 0 and x = 2. (d) Compute 2 f(x). How does this compare to the estimates in c)? 0 (e) Partition the interval [0, 2] into subintervals of length 1 and repeat parts c) and 4 d) for this new partition. What do you notice? What conclusions can be made? (f) Find the exact area bounded by f and the x-axis over the interval [0, 2]. 2. In each of the following sketch a graph(s) with the properties: (a) f(x) is not constant 0 and a f(x) dx = 0. a (b) f(x) < 0 for x ( 2, 1) and 0 f(x) dx > 0. 3 (c) f(x), g(x) < 0 x and 2 (f(x) g(x)) dx > 0. 1 (d) 2π f(x) dx = 0, 2π f(x) dx = 2 π f(x) dx Calculate the following derivatives using the chain rule and the fundamental theorem of calculus. a) ds dx c) ds dx x 0 x 2 x 2 dt b) ds x t 2 dx dt 1 + t 2 d) d2 dx 2 0 x 0 dt 1 + t 2 dt 1 + t 2 4. Let F (x) = x dt, x > 0. For the following do not use the fact that you know 1 t another name for this function. (a) Find F (x) and determine intervals where it is increasing and decreasing. (b) Find F (x) and determine the concavity of F (x). (c) Graph F (x).

8 LiCMATH Math 31 Calculus II Worksheet 6 1. Consider the series (a) Find the sum of the first 5 and the first 10 terms in the series. (b) Does the series converge? If so, what the sum of the series? 2. Which of the following series are geometric? If they are geometric, find the constant a and the common ratio r. If they are not geometric, explain why. (a) 2 + 2a + 2a 2 + 2a (b) 2 + 4a + 8a (c) 2 + 2ak + 2a 2 k 2 + 2a 3 k A ball is dropped from a height of 14 feet and bounces. Each bounce is 2 3 height of the previous bounce. of the (a) Find an expression for the height to which the ball rises after it hits the floor the nth time. (b) Find the total vertical distance the ball has travelled when it hits the floor the 4th time. (c) If the ball was to bounce forever, would it travel a finite distance? 4. A tennis ball is dropped from a height of 40 feet and bounces. Each bounce is 1/2 the height of the previous bounce. A superball has a bounce of 3/4 the height of a previous bounce, and is dropped from a height of 30 feet. Which ball bounces a greater vertical distance? 5. Find the sum of the series 8 n=1. (hint: The series is telescoping.) n 3 +3n 2 +n 6. Not all telescoping series converge. Consider n=1 n (n + 1). (a) Show the series diverges by expanding the partial sum s k and collapsing. (b) Show the series diverges by simplifying a n = n (n + 1). 7. Show the harmonic series diverges using the integral test.