Problems (The Derivative) 1 A rock is dropped from a height of 400 feet and falls toward the earth in a straight line. In t seconds, the rock drops a distance of s = 16t2 feet. What is the average velocity of the rock while it is falling? Use limits to find the instantaneous velocity of the rock when it hits the ground. 2. Given f (x )  x 2  6 , find the slope of the graph of f at the x-value x0 = 2. 3. Find the equation of the tangent line to y = f(x) = 3x at x = 4. 4. Use the definition of the derivative to calculate f (x) if f(x) = 3x4 + 9 and find the equations of the tangent line and the normal line to the graph of f at x = 3. –3x 3  8x 2  x dy Find if y  . x dx 6. Find equations for the tangents to the graph of y = 28 - 3x - x2 at those points where the curve intersects the x-axis.

5.

7. Find the points on the graph of y = 2x3 + 18x2 + 30x + 24 at which the tangent is parallel to the x-axis. 8. d 2y If y = x7 + 4x5, find . dx 2 9. d 2y 5 1 Find if y    2 . 2 t t dt 10. Find f (x) if f(x) = x cot x. 11. Find f (x) if f(x) = x cos x. 12. cos x dy Find if y  3 . dx x 13. dy Find if y  sec x tan x . dx 14. cot x Find f (x) if f (x )  . 1  csc x

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15. 16.

Find y(x) if y  7sin x  2cos x 

x7 . 7

1 . 1  4cos  17. 1  cos  Find f () if f ( )  . 1  cos  18. dy 1  tan t Find if y  . dt 1  tan t

Find f () if f ( ) 

19. f(x) = x2 cos x + sin x. Find f (x). 20. f(x) = 6x cot x. Find f (x). 21.

f (x ) 

sec x

. Find f '(x). x7 22. Find f (x) where f (x )  csc4 5x . 23. Find f (x) where f (x )  csc2 7x  x 2

.

24. Find f (x) where f(x) = (x3 - 6)7(x3 + 3)6. 1 25. 1  Find y (/2) where y   – cos x  . x 

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