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The Derivative – Day 1 WKST Name:

The Formal Limit Definition of the Derivative

Graphical and Numerical Understanding of the Derivative as a Slope and Rate of Change

1. Find the derivative, [pic], of [pic] 2. Verify the derivative, [pic], of [pic]

algebraically using the limit definition. in #1 by using the short-cut formula.

3. If [pic], then find [pic] algebraically by 4. Verify[pic]in #3 using the short-cut formula.

using the limit definition.

5. If [pic], then find [pic] numerically by 6. Verify the value of[pic]in #5 by

creating a table of values near [pic]. using the calculator’s Math 8 function.

Create a table to find the slope of each function at x = c. DO NOT USE THE SHORTCUTS.

7. [pic] 8. [pic]

at x = 1 at x = 2

YOU MAY USE THE SHORTCUT FORMULA FOR THE REST OF THE WORKSHEET (AND THE REST OF YOUR LIFE) WHEN IT'S APPLICABLE. IF YOU DO NOT KNOW THE SHORTCUT, THEN COMPUTE THE DERIVATIVE NUMERICALLY.

9. The difference quotient of [pic] 10. If [pic], then [pic] is

from x = 1 to x = 3 is

(A) 19 (A) 19

(B) 18 (B) 18

(C) 0 (C) 0

(D) -18 (D) -18

(E) -19 (E) -19

11. Draw the graph of [pic] and represent the answers from #9 and #10 on the graph.

12. The slope of the tangent line of [pic] 13. The slope of the secant line of [pic]

at [pic] is from [pic] to [pic] is

(A) 2 (A) 2

(B) 1 (B) 1

(C) 0 (C) 0

(D) -1 (D) -1

(E) -2 (E) -2

14. The average slope of [pic] 15. The derivative of [pic]

on the interval [pic]is at 3 is

(A) [pic] (A) [pic]

(B) [pic] (B) [pic]

(C) 0 (C) 0

(D) [pic] (D) [pic]

(E) [pic] (E) [pic]

16. The instantaneous rate of change of [pic] 17. The average rate of change of [pic]

at [pic] is on the interval [ 0 , 2 ] is

(A) [pic] (A) [pic]

(B) [pic] (B) [pic]

(C) [pic] (C) [pic]

(D) [pic] (D) [pic]

(E) [pic] (E) [pic]

18. Find the equation of the tangent line to [pic] at [pic].

(Hint: First find the value of f (t) at [pic], then find the slope at [pic], then use your algebra skills to find the equation in the form [pic]. Either use the old school formula, [pic], or substitute x , y, and m, into the equation [pic]and solve for b.)

Use the table of values for [pic]to answer #19 to #21 .

|[pic] |0 |1 |2 |3 |4 |

|[pic] |0 |2 |6 |12 |20 |

19. Find the best approximation for[pic] 20. Find the best approximation for[pic]

21. Find the best approximation for[pic]

22. If [pic] is a point on [pic]and the line tangent at that point passes through ( -2 , 4 ), then [pic]

(A) 1 (B) [pic] (C) -2 (D) [pic] (E) 3

23. Find [pic] if [pic]contains the points ( 2 , 1 ) and ( 4 , -3 ).

(A) 0 (B) [pic] (C) -2 (D) [pic] (E) 4

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ANSWERS:

1) [pic] 7) [pic] 13) C 19) 3

2) [pic] 8) 4 14) E 20) 6

3) [pic] 9) A 15) A 21) 8

4) [pic] 10) A 16) E 22) B

5) [pic] 11) graph 17) A 23) C

6) [pic] 12) D 18) [pic]

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