# Mathematics Practices

Exploration 15-4b: Limits and Curved Asymptotes

Objective: Analyze graphs of rational algebraic functions with the help of synthetic
substitution and factoring .

 1. Let Plot the graph of f as . Use a friendly window with an x-range of about to that includes as a grid point. Have the grid off. Sketch the graph. 2. is undefined because of division by zero. Trace to on your graph. What feature does the graph have at this point? By tracing closer and closer to , find the limit seems to be approaching as x approaches 1. Is the limit as x approaches 1 from the left side the same as when x approaches 1 from the right side? 3. Remove the discontinuity at algebraically by factoring the numerator and reducing the fraction . Evaluate the resulting quotient polynomial at Is the answer equal to the limit you found in Problem 2? 4. How do you read What does this equation mean? 5. Let Plot the graph of g as using “thick” style. What feature does the graph of g have at Sketch the graphs of and here, showing their relationship to each other. 6. Try to find by tracing to x- values closer and closer to 1. Try x-values on both sides of 1. What happens to the quotient as x approaches 1? 7. To understand why the graph of resembles the graph of simplify the equation for by synthetic substitution . Write the equation in “mixed- number ” form as What relationship do you notice between the equations for f (x) and g(x)? 8. Let Plot the graph of as . 9. The graphs of f and g from Problem 5 should look like this . On this figure, sketch the graph of h. 10. Simplify the equation for by long division or synthetic substitution. Write the result in mixed-number form as 11. How is the polynomial part of in Problem 10 related to the graph of ? 12. Why is it important for your work to be 100% correct in problems like these that are sequential? 13. Write a paragraph summarizing the things you have learned about rational functions, removable discontinuities, asymptotes, etc. as a result of doing this Exploration.

Exploration 15-5a: Rate of Change Date:
of a Polynomial Function

Objective: Find the instantaneous rate of change of a polynomial function at a given
value of x.

 Astronaut Spencer Spacey takes off from the planet Alderaan. He starts his rocket countdown at time minutes. Shortly thereafter his spaceship takes off. It rises for a while, then drops while his second-stage rocket engine is starting up, and then rises again. Spencer’s computer finds that his distance, miles, above the surface is given by 1. How far is Spencer from the surface at min? at min? How far did he go in this time interval? What was his average velocity for this time interval? 2. What is Spencer’s average velocity for the time interval from 2 minutes to x = 2.001 minutes? 3. Spencer’s instantaneous velocity at 2 minutes is the limit of his average velocity between 2 and x minutes as x approaches 2. From your answers to Problems 1 and 2, what do you conjecture that his instantaneous velocity equals at 2 minutes?4. Between time 2 and x minutes, Spencer travels miles. So his average rate is Substitute for and and simplify the numerator. Then use synthetic substitution to do the division, thus removing the discontinuity at Write a simplified equation for . 5. Use this simplified equation for to find the limit as x approaches 2. Does this confirm the instantaneous velocity you found in Problem 3? 6. Plot a line through the point that has a slope equal to the instantaneous velocity you found in Problem 5. Take into account that the two axes have different scales . How does the line relate to the graph? Relationship: 7. Find Spencer’s instantaneous velocity at time 3 minutes by starting with Explain the fact that the answer is negative . 8. Let . Find and . How do your answers compare with the instantaneous velocities at times and 9. There is an algebraic way to find the instantaneous rate of change of a polynomial function. Suppose that If is the average rate of change between the fixed time c and the variable time x, then Divide by means of synthetic substitution. Write a polynomial equation for . (It will involve the constant c.) 10. By appropriate calculations, show that 11. Explain how you can get the 5 and the 4 in from the original polynomial 12. Suppose that What do you think would be an equation for the function that gives the instantaneous velocity of ? 13. Show that you can use the patterns you found in Problems 11 and 12 on the function to get the velocity function from Problem 8, 14. Look in Section 15-5 to find the special name given to the velocity function. 15. What did you learn as a result of doing this Exploration that you did not know before?
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