Mid Term Pre Calc Review

Transcription

1 Mid Term Pre Calc Review I. Quadratic Functions a. Solve by quadratic formula, completing the square, or factoring b. Find the vertex c. Find the axis of symmetry d. Graph the quadratic function e. Evaluate complex number operations, add, subtract, multiply, divide f. Graph complex numbers on the complex plane (a+bi) II. Limits a. Define a limit b. Define the three types of discontinuity i. Point ii. Jump iii. Infinite c. Know the three ways of evaluating limits i. Direct substitution ii. Quotient method iii. Conjugate method III. Functions a. Determine if a function is either: i. Odd ii. Even iii. Symmetric about the x axis b. Determine the domain and range of a function c. Parent functions i. Know the parent functions as discussed in class ii. Know the transformations of functions given y=a(x-h) n + k iii. Determine the transformations of the following function: f(x)=-0.25(3x+2) 3 5 d. Function operations i. Add ii. Subtract iii. Multiply iv. Divide 1. Long division 2. Synthetic division only when denominator is in form (x-h) e. Polynomial functions i. End behavior as x approaches infinity, what happens to the graph? ii. Extrema 1. Local max and min 2. Global max and min 3. Point of inflection a. Be able to find the point of inflection of a given function Midterm Review Page 1 of 20

2 IV. Graphing Rational functions a. Find the existing asymptotes b. Find the existing intercepts c. How many times can a curve cross the horizontal asymptote? Once d. How many times can a curve cross the vertical asymptote? NONE e. Know the three cases of horizontal asymptote i. Be able to calculate the horizontal asymptote by evaluating the limit to infinity ii. As x approaches infinity if the limit is a number, there is a horizontal asymptote iii. As x approaches a number if the limit is infinity, there is a vertical asymptote f. Know the only case for oblique asymptote and how to calculate it g. Know the special case in which a possible point of discontinuity exists V. Calculate the inverse of a function a. Graph the function and its inverse VI. Calculate Average Rate of change a. Know that the slope of the secant line represents the aroc b. Calculate the equation of a secant line given an interval c. Calculate average velocity and graph a function with respect to time VII. Calculate Instantaneous rate of change a. Know that the slope of the tangent line represents the IROC b. Differentiate a function at a given point c. Determine if a function is differentiable at a given point d. Calculate the derivative of a function using either the power rule or difference quotient e. Calculate the equation of a tangent line given an exact point or x value f. Calculate the instantaneous velocity and graph a function with respect to time VIII. Graphing polynomial functions using the first derivative test a. Find the real zeroes of a polynomial function b. Determine the critical points of the graph using the first derivative test c. Determine the extrema using the second derivative test d. Determine the concavity and point of inflection using the derivative test e. Graph the function using the values/results of the derivative test and real zeroes IX. You will have up to 10 short answer questions a. Example: Describe what a single sided and double sided limit is? b. Example: Horizontal shift left 8 units, vertical shift up 1 unit, vertical compression 2/3 units, an x axis reflection of the square root parent function X. You will have two constructed response in which you will have to graph a rational function by evaluating limits and determine instantaneous velocity of an object Midterm Review Page 2 of 20

3 Practice problems: Solve the following quadratic functions by completing the square 1. g(x) = x 2 + 6x + 8 f(x) = -x 2 + 6x 7 c(a) = 1/2x Find the critical information for the following functions using any method you deem necessary: 2. c(h) = -(x + 5) 2-20 a. vertex b. axis of symmetry c. max or min value d. x intercept e. y intercept 3. x 2-12x + 20 = y a. vertex b. axis of symmetry c. max or min value d. x intercept e. y intercept Graph the following functions using any method: 4. y = x a. vertex b. axis of symmetry c. max or min value d. x intercept e. y intercept 5. y = -(2x - 4) 2-6 a. vertex b. axis of symmetry c. max or min value d. x intercept e. y intercept Midterm Review Page 3 of 20

4 Complete the following operations given a = i b = 3 - i: 6. (a+b) 7. b a 8. b/a 9. Graphically find the absolute value of the resultant vector of a and b. Simplify the following radicals Short answer/constructed response: 11. Explain why this relation is not a function. x = [-1,0,2,3,-1] y = [1,2,3,4,5] 12. The following function represents the path of a baseball thrown; h(t) = -9.8t 2 + v0t +4 where v0 = initial velocity in m/s. Complete the following items below; a. If Beau throws the ball with an initial velocity of 90m/s, write a function h(t) modeling the path that the ball follows. b. Given your function h(t), how high is the ball above the ground after 1 second? c. Graph the path of Beau's pitch. Be sure to correctly label your independent and dependent variable on the proper axes Midterm Review Page 4 of 20

5 d. After how many seconds does the ball reach its maximum height? e. What is the balls maximum height? Fill in the blank 1. An even function is symmetric about the axis. 2. An odd function is symmetric about the axis. 3. The mathematical test to determine if a function is even or odd is to replace with 4. If for every (x,y) on a graph there exists an (x,-y), then the graph is symmetric with respect to the. 5. If for every (x,y) on a graph there exists an (-x,-y), then the graph is symmetric with respect to the Determine if the following functions are Even, Odd, or Neither either mathematically or visually f(x)= 3x f(x) = x f(x)= 3x Midterm Review Page 5 of 20

6 15. Find the domain and range of the following graphs in interval notation. Use a union symbol if needed. Domain Range Domain Range 16. Use the following function to answer the below items: g(x) = x 2 ; x 3 a. Sketch a graph of g(x). b. What type of discontinuity does g(x) exibit? Midterm Review Page 6 of 20

7 c. At what point is g(x) discontinuous? d. Write a statement of discontinuity using the above information e. What is the domain of g(x) in interval notation? Use a union if needed. f. What is the range of g(x) in interval notation? Use a union if needed. g. What is the limit of g(x) as x approaches 2 from the left and the right? Evaluate the limit using the appropriate method: Show your work on here! Circle final answer! 1) lim ( (3+x) 2 9)/(x) = x 0 2) lim (x+6)/(x x + 24) = x - 6 3) lim (x-49) / ( x + 7) = x 4 9 4) lim ½x 2 4x + 6 = x -6 5) Lim (3x 2 + 5x + 6) / (x+1) = x -1 6) Lim (-3x) / (x 1) = x Midterm Review Page 7 of 20

8 7) What type of discontinuity occurs when the left hand limit is equal to the right hand limit? 8) What type of discontinuity occurs when the left or right hand limit is equal ±? Parent Function F(x) = x 2 Sketch a Graph Domain and Range End Behavior x + Name x - F(x) = x 3 x + Name x - F(x) = x x + Name x - F(x) = x x + Name x Midterm Review Page 8 of 20

9 F(x) = a x x + Name x - F(x)= 1/x Name x + x - F(x) = 1/x 2 x + Name x - F(x) = x x + Name x Midterm Review Page 9 of 20

10 Behavior of graphs: 1. Given the following graph, find the: (round to nearest 10 th ) a. Local maximum b. Relative minimum c. Global minimum d. Absolute maximum e. Range 2. Given the function, g(x) = - (x + 3) 3 + 2x 1, sketch the graph and identify the following to the nearest 10 th : a. Local max b. Local min c. Point of Inflection d. Domain 3. Graph the function h(t) = (t-2) 3 4x 4 ; x 0 sketch the graph and identify the following to the nearest 10 th : a. Local max b. Local min c. Global min d. Point of Inflection e. Domain Midterm Review Page 10 of 20

11 Answer the following questions. Round all answers to the nearest 10 th. READ THE QUESTIONS!!! Answer the following as discussed in class 4. What is the point of Inflection? (Not WHERE is it!) 5. How does the absolute value of the a term being greater than one affect a graph? 6. If the graph is concave down on the left of x and concave down on the right of x, is there a maximum or minimum value at x. (Circle one) Transformations of parent functions 7. Given the form of a function below, describe each transformation as discussed in class: F(x) = af(x-h) + k a. negative a term b. absolute value of the a term between 0 and 1 c. absolute value of the coefficient of x d. the h term e. the k term 8. Given g(x) = (2x + 6) 2 8, fill in the blanks: a. The negative (a) term indicates a axis b. The horizontal translation is units to the c. The vertical translation is units d. Because the absolute value of the (a) term is 0.66, there is a compression Use the description of the transformation to construct the function 9. Horizontal shift left 8 units, vertical shift up 1 unit, vertical compression 2/3 units, an x axis reflection of the square root parent function Midterm Review Page 11 of 20

12 10. Horizontal shift right 2 units and a vertical shift up 1 unit of the reciprocal parent function 11. Given the graph, below write the function based on the transformations: (assume no vertical or horizontal compression or stretch) Behavior of graphs 12. Given the following graph, find the: (round to nearest 0.5) a. Local maximum b. Relative minimum c. Global minimum d. Absolute maximum e. Range Midterm Review Page 12 of 20

13 Part I: Function operations Given a(x) = (-2x + 1), f(x) = 4x 2 + 3x +2 and g(x) = 2x 2 5x 6, calculate the following: 1. (f+g)(x) 2. (g f)(x) 3. (ag)(x) 4. (fa)(x) 5. True or false: Division of polynomials is commutative? 6. Put f(x)= -2x + 4x x 2 into polynomial form. Part II: Function inverse 7. How do you calculate the inverse of a function? 8. Are all inverse relations functions? 9. The domain of h(x) is the of its inverse and the range of h(x) is the of its inverse. 10. In the case f(x)=f -1 (x), it is called an involution. Is g(x) an involution of f(x) if f(x) = -x-1 and g(x) = -x 1. Show how you determined this. 11. Calculate the inverse of h(x) = -2x Given f(x) = (-4)/(x+2), calculate f -1 (x). 13. Determine if f(n) is the inverse of g(n) using a table of values Calculate the inverse of each and graph the following functions and their inverse on the graph provided: Midterm Review Page 13 of 20

14 Part III: Polynomial Division 16. Use synthetic division to evaluate the following: a. (3x 2 + 7x +2)/(x+2) b. (3x x 3 )/(x-1) 17. Use long division to evaluate the following: a. (x 2 + 6x + 8x)/(-x-1) b. (-6x + 9 3x 3 )/(-3x + 1) Part IV: Compound Functions: Complete the following Short answer/fill in the blank 1. If the degree of the numerator is greater than the degree of the denominator, the Horizontal Asymptote is 2. If the degree of the numerator is equal to the degree of the denominator, the Horizontal Asymptote is 3. If the degree of the numerator is less than the degree of the denominator, the Horizontal Asymptote is 4. The oblique asymptote exists ONLY when the degree of the numerator is than the degree of the denominator. Graphing/Calculating asymptotes and intercepts: 5. Calculate the following asymptotes if a = x 2 + 9x + 20 b = x 2 6x + 40 c = x + 1 d = 2x + 4: a. G(x) = 1/c i. Vertical Asymptote ii. HA iii. OA iv. Y intercept v. X intercept vi. Graph g(x) on the graph provided Midterm Review Page 14 of 20

15 6. Given g(x) = (2x 3 5x 2 + 4x 1)/(x-1), the point of discontinuity is: a. (1,0) b. (-1,6) c. (6,-1) d. (0,1) e. DNE 7. Given y = (3x-6)/(2x+1), the x intercept is: a. 6 b. -6 c. 2 d If f(x) = (25x 2 + 6)/(8x-5x 2 + 4), the Horizontal Asymptote is: a. 5 b. -5 c. DNE d Given h(x) = (3x 2 + 4x + 1)/(2x x + 24), the vertical asymptote is/are: a. X = -2, x = -6 b. X = 2, x = 6 c. VA DNE d. X If f(x) = -2/(3x ), the limit as x approaches infinity is? a. DNE b. Double Sided c. 0 d. -2/ Midterm Review Page 15 of 20

16 Graphing/Calculating Rational functions: 11. Calculate the following asymptotes if a = x 2 + 9x + 20 b = x 2 6x + 40 c = x + 1 d = 2x + 4: a. G(x) = 1/c i. Vertical Asymptote ii. HA iii. OA iv. Y intercept v. X intercept vi. Graph g(x) on the graph provided b. F(x) = 3/d i. Vertical Asymptote ii. HA iii. OA iv. Y intercept v. X intercept vi. Graph f(x) on the graph provided c. H(x) = a/c i. Vertical Asymptote ii. HA iii. OA iv. Y intercept v. X intercept Midterm Review Page 16 of 20

17 Calculate the following asymptotes if a = x 2 + 9x + 20 b = x 2 6x 40 c = x + 1 d = 2x + 4 d. C(x) = b/(x+4) i. Vertical Asymptote ii. HA iii. OA iv. Y intercept v. X intercept vi. Point of discontinuity e. g(x) = (2x 3 5x 2 + 4x 1)/(x-1) i. Y intercept i. X intercept ii. Vertex iii. Point of discontinuity Midterm Review Page 17 of 20

18 Derivatives and differentiation: Differentiate the following using the power rule: 1. G(x) = x 2 2.) c(t) = t 3 3) g(x) = -2x 5 4.) c(x) = -0.25x 4 6x 2 + 8x 4 Differentiate the following for the values given: 5. q(t) = -16t 2 + 9t 1 for x = 1 6.) c(d) = -x 5 + 6x 2 3 for x= The slope of the tangent line represents what? 8. What does the derivative of a function represent? Be very specific!!! Equations of tangent lines: 9. Given the function h(t) = -t 2 + 2t 1, find the equation to the line tangent to h(t) at t=(-1). 10. Given the function c(x) = -x 3 4x + 2, find the slope of the tangent line at the following points: a. (0, 2) b. (1,-3) Instantaneous Rates of Change Applications. Sketch a graph of the scenario and answer the questions: 11. The function M(t) = -t 2 + 6t + 2, 0 t 6 represents the path of a particle in a vacuum over a time interval of 1 to 5 seconds, where t = seconds and M(t) represents the vertical displacement (distance from 0) in miles. Round all answers to the 1000 th. a. Sketch a graph to model the M(t) over the time interval of 0 to 6 seconds. (Label the axes) Midterm Review Page 18 of 20

19 b. Where is the particle s initial starting point at time 0? c. What is the particles average velocity between 3 and 6 seconds? Why is the velocity negative? d. What is the particles instantaneous velocity at 0 seconds, 3 seconds, and at 6 seconds? e. Form a conjecture as to why the particles velocity at t=3 0 miles per second? Why would the velocities for 0 and 6 seconds be the same? 12. A proton travels in a vacuum along the path modeled by D(t) = 2t 2 1 where t is time in seconds and D(t) is displacement in miles. a. Graph D(t), label the axes correctly! b. What is the particle s instantaneous velocity at 0 seconds Midterm Review Page 19 of 20

20 c. What is the particle s instantaneous velocity at 10 seconds d. Graph a tangent line to D(t) at 1 second Midterm Review Page 20 of 20