MTH 141 Final Exam Review

MTH 141 Final Exam Review

MTH 141 Final Exam Review

Evaluate the following limits:

3.)

4.)

5.) If f(x)=x3+4x, estimate f’(3) using a table with values of x near 3, spaced by 0.001.

6.) Using the “Limit Definition of the Derivative”, show that the derivative of the function f(x)=2x is 2.

7.) Using the “Limit Definition of the Derivative”, find the derivative of the function g(x)=2x2+3x-1 at x=1.

8.) In a time of t seconds, a particle moves a distance of meters from its starting point, where s=4t2+4. Find the average velocity between t=1 and t=1+h when h=0.1, 0.01 and 0.001 (3 separate answers). Use these answers to estimate the instantaneous velocity at t=1.

9.) Graph the f’(x) function on top of the f(x) functions to the right:

10.)

For problems 11-15, solve for f’(x):

  1. f(x)=x2e-2x+e-x
  1. f(x)=ln(2+e4x)
  1. f(x)=
  1. f(x)=sinh(x3)+cosh(2x)
  1. f(x)=sin(cos())
  1. x2+y2=16, find dy/dx
  1. sinh(r2)*eA=r+A, find dA/dr
  1. x3t5+3x=8t3+1, find dt/dx
  1. e2a+3b=a2-ln(ab3), find db/da
  1. g(t)=t4+2t2, find the tangent line to the graph at t=2
  1. h(u)=sin(2u)+u, find tangent line to the graph at u=
  1. (1+x)y=sin(xy2)- x, find tangent line to the graph at x=0
  1. Use local linear approximation to approximate the quantity (1.98)3
  1. Use local linear approximation to approximate the quantity
  1. Use local linear approximation to approximate the quantity
  1. f(x)=x3-(3/2)x2, find and classify critical points
  1. On the interval [-7,5], find the global max and min of the function f(x)=
  1. g(t)=, find and classify critical points
  1. Using optimization, find two positive numbers “x” and “y” where x+2y=76, and where the product x*y is the greatest.
  1. The length of a rectangular solid is twice its height. The width of the solid is y. What are the dimensions of the box with the maximum volume, given the surface area is 64 cm2 ?
  1. A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must 180,000 m2 in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

32.)A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is expanding at the rate of 4 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when its dimensions are 12 cm by 8 cm?

33.) Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.

34.) A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

35.) A tank of water in the shape of a cone is leaking water at a constant rate of . The base radius of the tank is 5 ft and the height of the tank is 14 ft.

(a) At what rate is the depth of the water in the tank changing when the depth of the water is 6 ft?

(b) At what rate is the radius of the top of the water in the tank changing when the depth of the water is 6 ft?

36.) A light is on the top of a 12 ft tall pole and a 5ft 6in tall person is walking away from the pole at a rate of 2 ft/sec.

(a) At what rate is the tip of the shadow moving away from the pole when the person is 25 ft from the pole?

(b) At what rate is the tip of the shadow moving away from the person when the person is 25 ft from the pole?

For Questions 37-40 use L’ Hospital’s Rule

41.) A particle moves in the x-y plane according to:

X(t) = 3t4 + 2t2 + 3 and y(t) = 2t2 – 3t

Find the speed of the particle at t = 2

42.) A particle moves in the x-y plane according to:

X(t) = t – sin(4t) and y(t) = cos(2t) + 5

Find the speed of the particle at t =π

43.) What curve do these parametric equations trace out? Find the Cartesian equation and graph out a few points.

X=3t+1

Y=t-4

44.) Find the equation of the tangent line to the curve at the given value t.

X=t3-t

y=t2

t=2

45.) a. ∫ab f(x)dx

b. ∫bc f(x)dx

c. ∫ac f(x)dx

d. ∫ac | f(x)| dx

46.)

47.) Find the average value of f(x)=2x2-2 for [1,4]

48.) Find the average value of g(t)=1+t for[0,2]

49.) Find the area of the region under the curve through use of the integral and the Fundamental Theorem of Calculus for the function f(t) within the interval [a,b].

f(t)=t2

a=1, b=3

50.) Find the area of the region under the curve through the use of the integral and Fundamental Theorem of Calculus for the function f(t) within the interval [a,b].

f(t)=1/t

a=1, b=e

In Problems 51-53, find the antiderivate F(x) is F’(x) is f(x) and F(0)=0.

51.) f(x)=3

52.) f(x)=2+4x+5x2

53.) f(t)=1/t

In Problems 54 and 54, find the indefinite integral:

54.) f(x)=x2+5x+8

55.) g(t)=4/(t2)

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