CALCULUS - Ed Kornberg



CALCULUS

WORKSHEET ON DERIVATIVES OF PRODUCTS AND QUOTIENTS

Work the following on notebook paper. Do not use your calculator.

Find the derivative. Do not leave negative exponents or complex fractions in your answers.

1. [pic] 3. [pic] 5. [pic]

2. [pic] 4. [pic]

6. Write the equation of the tangent line to the graph of [pic] at the point where [pic].

7. Write the equation of the tangent lines to the graph of [pic] that are

parallel to the line [pic].

8. Prove: [pic]

9. The length of a rectangle is given by 2t + 1, and its height is [pic], where t is the

time in seconds and the dimensions are in centimeters. Find the rate of change of

the area of the rectangle with respect to time.

10. Find a second-degree polynomial [pic] such that its graph has a tangent

line with slope 10 at the point (2, 7) and an x-intercept at (1, 0). Check your answer

with the given information.

11. (a) Given [pic]. Graph the function.

(b) If you “zoomed in” at x = 4 on your graph, what would the slope be from the left

side of x = 4? From the right side?

(c) Do you think f is differentiable at x = 4? Prove your conclusion by using the

alternative form of the definition of the derivative. Sketch a graph that helps you

evaluate the alternative form.

(d) Find [pic] by using the Power Rule. Is the graph of [pic] continuous at x = 4?

12. (a) Given [pic]. Graph the function.

(b) If you “zoomed in” at x = 0 on your graph, what would the slope be from the left

side of x = 0? From the right side?

(c) Do you think f is differentiable at x = 0? Prove your conclusion by using the

alternative form of the definition of the derivative.

(d) Find [pic] by using the Power Rule. Does [pic] exists?

(e) Sketch the graph of [pic]. Is the graph of [pic] continuous at x = 0.

13. (a) Given [pic]. Graph the function.

(b) If you “zoomed in” at x = 1 on your graph, what would the slope be from the left

side of x = 1? From the right side?

(c) Do you think f is differentiable at x = 1? Prove your conclusion by using the

alternative form of the definition of the derivative.

(d) Find [pic] by using the Power Rule. Does [pic] exists?

(e) Sketch the graph of [pic]. Is the graph of [pic] continuous at x = 1?

Find [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

25. [pic]

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