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Chapter 3 | Derivatives
3.1 EXERCISES For the following exercises, use Equation 3.1 to find the slope of the secant line between the values x 1 and x 2 for each function y = f ( x ).
19. f ( x ) = 2
a =−4
x +3 ,
20. f ( x ) = 3 x 2
, a =3
1. f ( x ) =4 x +7; x 1 =2, x 2 =5 2. f ( x ) =8 x −3; x 1 =−1, x 2 =3 3. f ( x ) = x 2 +2 x +1; x
For the following functions y = f ( x ), find f ′( a ) using Equation 3.1 .
21. f ( x ) =5 x +4, a =−1 22. f ( x ) =−7 x +1, a =3 23. f ( x ) = x 2 +9 x , a =2 24. f ( x ) =3 x 2 − x +2, a =1 25. f ( x ) = x , a =4 26. f ( x ) = x −2, a =6
1 =3, x 2 =3.5
4. f ( x ) =− x 2 + x +2; x
1 =0.5, x 2 =1.5
5. f ( x ) = 4
x 1 =1, x 2 =3
3 x −1 ;
6. f ( x ) = x −7
x 1 =0, x 2 =2
2 x +1 ;
7. f ( x ) = x ; x 1 =1, x 2 =16 8. f ( x ) = x −9; x 1 =10, x 2 =13 9. f ( x ) = x 1/3 +1; x 1 =0, x 2 =8 10. f ( x ) =6 x 2/3 +2 x 1/3 ; x
27. f ( x ) = 1 x , a =2
28. f ( x ) = 1
a =−1
x −3 ,
1 =1, x 2 =27
29. f ( x ) = 1 x 3
, a =1
For the following functions, a. use Equation 3.4 to find the slope of the tangent line m tan = f ′( a ), and b. find the equation of the tangent line to f at x = a . 11. f ( x ) =3−4 x , a =2 12. f ( x ) = x 5 +6, a =−1 13. f ( x ) = x 2 + x , a =1 14. f ( x ) =1− x − x 2 , a =0
30. f ( x ) = 1 x , a =4 For the following exercises, given the function y = f ( x ), a. find the slope of the secant line PQ for each point Q ⎛ ⎝ x , f ( x ) ⎞ ⎠ with x value given in the table. b. Use the answers from a. to estimate the value of the slope of the tangent line at P . c. Use the answer from b. to find the equation of the tangent line to f at point P .
15. f ( x ) = 7 x , a =3 16. f ( x ) = x +8, a =1 17. f ( x ) =2−3 x 2 , a =−2
18. f ( x ) = −3
a =4
x −1 ,
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