Practice for Midterm

 

Find the average rate of change of the function over the given interval. Round to nearest thousandth, if necessary.

1) y = 4- 6 + 8, [- 2, 3]

Complete the table for the function and find the indicated limit.

2) If f(x) = , find lim f(x) x→1

 

3) Use the graph to estimate the limit of the function.

(a) f(x) (b) f(x)

4)
Find f(x) and f(x) and f(x)

5) Evaluate the limit.

 

6)   

7)  

8) Write the definition of the derivative. Find the derivative of the given function at the given point using the definition. 

f(x) = 3 - 2 for a = -1

9) Find the limit, if it exists.

Let f(x) =

f(x)

10) Find all points where the function is discontinuous.

 

11) Find the x-values where a) the function is discontinuous b) the function is not differentiable.

 

Find the intervals on which the function is continuous.

12) y =

13) y =

14. Find the equation for the tangent to the curve at the given point.

f(x) = 5x2 + x; (-4, 76)

Solve the problem.

15. Find the points where the graph of the function have horizontal tangents.

f(x) = - 21x

16. Find an equation of the tangent to the curve f(x) = - 2x + 1 that has slope 2.

17. Use the definition to find the function's derivative. Then evaluate the derivative at the indicated point.  g(x) = 3x2 - 4x, (3)

18. Find the derivative. f(x) = 4x4 + 3x3 + 6

19. Find an equation for the line tangent to given curve at the indicated point.   y = x - at ( 2, -2)

20. The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds, for the given time interval. Use s(t) to find the indicated quantities.

s = - + 2 - 2t,       0 ≤ t ≤ 2

Find the body's velocity and acceleration at the end of the time interval.

21. A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point?Find the indicated derivative.

22. Find if y = + .

23. y =

24 y =
25.Find the tangent to the curve y = at the point (1, 12).

Find the derivative.

26  y = +

27. r = 18 - cos θ

28.Find the indicated derivative.

Find if y = 4 csc x.

29. Find the derivative of the function.

y = +

30. r =

31. f(θ) = sin cos

32. y = (π t - 13)

33. q = cos

34. h(x) =

35. y =

36. Find .

36. y =
37. y = 4 cot
Find dy/dx by implicit differentiation.  

38. x3 + 3x2y + y3 = 8

39. y = 4

40. Find the slope at the indicated point on the given curve.

= 64, slope at (2, 1)

Find the related rate equation.

41) Suppose that the radius r and volume V = π of a sphere are differentiable functions of t. How is dV/dt related to dr/dt?

42. A product sells by word of mouth. The company that produces the product has noticed that revenue from sales is given by where x is the number of units produced and sold. If the revenue keeps changing at a rate of per month, how fast is the rate of sales changing when 1400 units have been made and sold?

43.The radius of a right circular cylinder is increasing at the rate of , while the height is decreasing at the rate of . At what rate is the volume of the cylinder changing when the radius is 7 in. and the height is 16 in.?

44. Find the absolute extreme values of each function on the interval. 

F(x) = 3-16+18, -1 ≤ x ≤

4 5. Find the extreme values of the function and where they occur.

y =

46. Find each critical point and determine if it is a max, min or neither

y =

47. y =

Use analytic methods to find the intervals on which the function is increasing, decreasing, concave up, and concave down. Also, locate and identify the local extrema and inflection points.

48. y = - 27x

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