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AP Calculus AB Section 2.1-2.2 Worksheet Name ________________________ Date ______________Period _____ Find the slope of the line tangent to the graph of the function at the given point. (Use the limit process to find the derivative). 1. f ( x) 2 5x; 1,7 2, 4 2. g ( x) 8 x2 ; 4. f ( x) 5 x 2 5 x 8 Find the derivative of the function by the limit process. 3. h( x ) 5 6 x 7 Write the equation of the line tangent to the graph at the given point. (Shortcuts are acceptable). 5. f ( x) x2 7 x 4; 4, 8 6. f ( x) 2 x ; 9, 6 Match the graph of the function to its derivative. 7. Find the derivative of the function. 8. f ( x) x 3 9. f ( x) 1 x8 10. g ( x) x 5 5 x 3 11. f ( x) 6 x3 6 x 2 2 12. f ( x) 6 x 2 4cos x 13. f ( x) 3x3 2sin x 14. 2 f ( x) 7 5cos x x 15. x4 6 f ( x) x3 Determine all points, (if any), at which the graph of the function has a horizontal tangent. 16. y x3 12 x 2 1 17. y x4 4x 5 Answer the following questions. 18. A ball is thrown straight down from the top of a 210-ft building with an initial velocity of -24 ft per second. The position function is s (t ) 16t 2 vot so . What is the velocity of the ball after 1 second? 19. A projectile is shot upwards from the surface of the earth with an initial velocity of 148 meters per second. The position function is s (t ) 4.9t 2 vot so . What is its velocity after 4 seconds? 20. Find the average rate of change of y with respect to x on the interval 2, 2 , where y 3x 2 2 x . 21. Create a sign chart to show intervals on which the function is increasing/decreasing and on which the derivative is positive/negative. Sketch the graph of the derivative on the same set of axes as the function. a b Answers: 1. 2. -5 -4 6 7 4. 10 x 5 5. y 8 x 4 3. 6. y 6 9 x 9 7. A 8. f '( x) 3x 2 8 x9 10. g '( x) 5 x 4 15 x 2 9. f '( x) 11. f '( x) 18 x 2 12 x 12. f '( x) 12 x 4sin x 13. f '( x) 9 x 2 2cos x 2 14. f '( x) 7x 6 7 5sin x 18 x4 16. 0,1 and 8, 257 15. f '( x) 1 17. 1, 2 18. 19. 20. 21. -56 ft/s 108.8 m/s -2 Graph