Calculus Honors and Introduction to Calculus

Transcription

1 Calculus Honors and Introduction to Calculus Name: This summer packet is for students entering Calculus Honors or Introduction to Calculus in the Fall of 015. The material represents concepts and skills in Algebra and PreCalculus that students need to be successful in Calculus. Students are expected to complete this packet by the first day of school. We strongly suggest that students do not wait until the last minute to complete the packet. In Calculus they will have the opportunity to synthesize what they have learned in geometry and algebra to understand how to analyze the behavior of mathematical functions. Thoughtful preparation for Calculus class includes careful completion of this packet. Students are encouraged to work together and to bring unresolved questions to class. The packets will be collected and assessed. There will be opportunities in-class and after-school to have packet-inspired questions answered. An exam on the contents of this packet will be given during the first quarter. This packet covers the following topics: 1. Mathematical functions (lines, polynomials, rational functions, exponential functions, logarithms, trigonometric functions).. Algebraic operations: a. Composition of functions b. Multiplication of polynomials c. Operations with logarithms and exponents d. Simplification of rational expressions 3. Inverse functions 4. Equations of lines 5. Trigonometric Identities 6. Roots of polynomials 7. Transformations on functions; the impact of algebraic changes on function graphs 8. Important: You are being asked to analyze some familiar functions in new ways. These problems are very important for you to try. 1

2 In Calculus we study the behavior of functions, examining how quickly or slowly functions change, where the function is increasing or decreasing and where the function changes direction. An important application in calculus is Distance, Velocity and Acceleration, where we use functions to describe these quantities. Even though you don t yet know calculus, you probably do understand the relationship between distance, velocity and acceleration (velocity describes how distance changes (e.g. miles per hour) and acceleration describes how velocity changes. Use your understanding of these ideas to answer the questions in section 1: I. Behavior of Functions 1. The distance travelled by an ordinary car, in the first 5 seconds of motion, is shown in the chart below. t seconds d feet a. Use this data to graph distance as a function of time in seconds. Use your own graph paper or print out the graph paper provided on chs/math. Make sure that time is on the horizontal axis and distance is on the vertical axis (usually the x-axis and y-axis, respectively.) b. Use the graph to estimate the distance travelled between: i. 3.5 and 4 seconds ii. iii. 4.5 and 5 seconds 5 and 6 seconds

3 . A ball is thrown in the air with an initial velocity of 40 ft / s. The function s( t) 40t t tells us how high the ball is after t seconds. a. Graph this distance function from t 0 to t 40 seconds. Make sure that t is graphed on the horizontal axis (usually called the x-axis) and that st () is on the vertical axis (usually called the y-axis). Use your own graph paper or print out the graph paper posted on chs/math. b. Using the graph, describe the velocity of the ball and how it changes. Indicate whether you think that it is increasing or decreasing. Put your observations in the chart below: Time Frame in seconds t 0 to t 1 Velocity: Is it Increasing or Decreasing? Explain why you think the velocity is increasing or decreasing. t 5 to t 10 t 1 to t 18 t 19 to t 1 t to t 8 t 30 to t 35 t 39 to t 40 3

4 3. A certain type of bacteria is known to double in population every 3 hours (if living conditions are good). Suppose the initial population is 100 bacteria. a. What is the size of the population after 15 hours? b. What is the size of the population after t hours? c. Estimate the size of the population after 0 hours. d. Graph the population function (using your own graph paper or the graph paper you can print from chs/math) and estimate how long it takes for the bacteria population to grow to 50,000. 4

5 II. More About Functions 4. How do determine whether or not a graph represents a mathematical function? 5. What is the inverse of the function y x 9? a. Is this inverse also a function? Why or why not? 6. Find the ( xy, ) coordinates for 3 points on each: a. y x 7 b. y7 x c. y e x sin( x) III. Writing equations of lines 7. Write the equation of the line that has a slope of 5 and goes through the point (1, ) 5

6 8. Write the equation of the line that connects ( 3,1) and ( 1, 4). 9. Write the equation of the line that is perpendicular to x3y10 and goes through the point (0,0) 10. Write the equation of a line that connects any point on the function y x to (5,5). IV. Rational Expressions Simplify the following expressions: x x1 1. x y y x x x 6

7 13. ( x h) x h 14. (( x h ) 3) ( x 3) h V. Remember quadratic functions Multiply the binomials and simplify: 15. ( x )(3x 5) 16. (1 x)( x1) 17. x( x 1)( x 3) VI. Factor the expressions to find all the roots of the functions below 18. y x x 19. f t ( ) 9t g( z) 1z 7

8 1. p x x x x ( ) p x x x x x ( ) VII. Trigonometric Functions 3. Simplify 1 sin ( x) cos ( x) 4. What are the roots of the function: f ( x) sin( x) 5. What are the roots of the function: g( x) sin( x) cos( x) 6. Why is the domain of the function h( x) tan( x) different from the domain of the function f ( x) sin( x)? Explain your answer 7. Graph these functions and, for each transformation on the parent function, y sin( x), explain why the graph has changed. Use graph paper and be as precise as you can. a. y sin( x) b. y sin(3 x) c. y 3sin( x) d. y 3sin( x) 1 8

9 e. y sin( x ) VIII. Composition of Functions 8. Given the graph below, evaluate each expression i. g( f( )) ii. ( f g)(0) iii. ( g f)(1) iv. ( f f)() v. ( g f)( 4) vi. Using the same graph you used in #1, explain the domain and range of each function ( f( x) and gx ( )). 9

10 Compose the following functions as indicated in each problem: 1 f ( x) x 1 gx ( ) h( x) sin( x) k( x) x x 9. f g( x) 30. g f ( x) 31. g g( x) 3. f f ( x) 33. f ( h( x)) 34. h( k( x)) 35. h( k( f ( x))) 36. f ( h( k( x))) 37. g( f ( h( k( x)))) 10