question archive 1) Verify that Rolle's Theorem can be applied to the function f(x) = x3-8x2+19x-12 on the interval [1,4]
1) Verify that Rolle's Theorem can be applied to the function f(x) = x3-8x2+19x-12 on the interval [1,4]
Subject:MathPrice: Bought3
1) Verify that Rolle's Theorem can be applied to the function f(x) = x3-8x2+19x-12 on the interval [1,4]. Then find all values of c in the interval such as f'(c)=0. Enter the exact answers in increasing order.
c=
c=
2. Verify that the Mean Value Theorem can be applied to the function f(x) = x4/5 on the interval [0,1]. Then find the value of c in the interval that satisfies the conclusion of the Mean Value Theorem.
c=
3. Consider the function f(x)=x−1/x+3
(a) Find the domain of f(x)
Note: Use the letter U for union.
Domain:
(b) Give the horizontal and vertical asymptotes of f(x), if any.
Enter the equations for the asymptotes. If there is no horizontal or vertical asymptote, enter NA in the associated response area.
Horizontal asymptote:
Vertical asymptote:
(c) Give the intervals of increase and decrease of f(x)
Note: Use the letter U for union.
Increasing:
Decreasing:
(d)Give the local maximum and minimum values of f(x).
Enter your answers in increasing order of the xx-value. If there are less than two local extrema, enter NA in the remaining response areas.
Include a multiplication sign between symbols. For example, a⋅π.
f(__)=__ , (Local maximum, local minimum, NA)
f(__)=__ , (Local maximum, local minimum, NA)
(e) Give the intervals of concavity of f(x).
Concave upward:
Concave downward:
(f) Give the inflection points of f(x).
Enter your answers in increasing order of the xx-coordinate. If there are less than two points of inflection, enter NA in the remaining response areas.
(_,_)
(_,_)
(g) draw the graph of f(x)
4. Consider the function f(x) = x + 15x2/3
a) Find the domain of f(x)
Note: Use the letter U for union.
Domain:
(b) Give the horizontal and vertical asymptotes of f(x), if any.
Enter the equations for the asymptotes. If there is no horizontal or vertical asymptote, enter NA in the associated response area.
Horizontal asymptote:
Vertical asymptote:
(c) Give the intervals of increase and decrease of f(x)
Note: Use the letter U for union.
Increasing:
Decreasing:
(d)Give the local maximum and minimum values of f(x).
Enter your answers in increasing order of the xx-value. If there are less than two local extrema, enter NA in the remaining response areas.
Include a multiplication sign between symbols. For example, a⋅π.
f(__)=__ , (Local maximum, local minimum, NA)
f(__)=__ , (Local maximum, local minimum, NA)
(e) Give the intervals of concavity of f(x).
Concave upward:
Concave downward:
(f) Give the inflection points of f(x).
Enter your answers in increasing order of the xx-coordinate. If there are less than two points of inflection, enter NA in the remaining response areas.
(_,_)
(_,_)
(g) draw the graph of f(x)
5. Show a graph which satisfies all of the given conditions. Justify your answer in terms of derivatives and concavity information below. You should explain why the graph you chose is correct as opposed to a solution by eliminating options. Specifically, your explanation should be a guide for how to construct the appropriate graph given only the information below and not the answer choices.
f'(-3)=f'(2)=0
Increasing on (-3, 2)U(2, infinity) decreasing (-infinity,-3)
f''(-4/3)=f''(2)=0
Concave upward
(-infinity, -4/3) concave downward (-4/3, 2)
6. Find the linearization of the function f(x) = 1/6x+5 at x = -1
L(x)=_____
Find the true values of f(x) and the approximations using L(x) below. Round all of your answers to 3 decimal places. You can use a calculator, spreadsheet, browser, etc. to calculate the exact values.
| x | The exact value f(x) | The approximate value L(x) |
| -1.1 | ___________________ | ___________________ |
| -1.01 | ___________________ | ___________________ |
| -1.001 | ___________________ | ___________________ |
7. Compute the values of dy and Δy for the function y=e3x+6x given x=0 and Δx=dx=0.02.
Round your answers to four decimal places, if required. You can use a calculator, spreadsheet, browser, etc. to calculate dy and Δy.
dy=______
Δy=______
8.Use differentials to estimate the value 4√16.4. Compare the answer to the exact value of 4√16.4. Round your answers to six decimal places, if required. You can use a calculator, spreadsheet, browser, etc. to calculate the exact value.
estimate=_________
exact value=_________
9. The radius of a circle is increasing at a rate of 10 centimeters per minute. Find the rate of change of the area when the radius is 3 centimeters. Round your answer to one decimal place.
The rate of change of the area is (number here) (units here)
10. Assume that xx and yy are both differentiable functions of tt and are related by the equation y=cos(5x).
Find dy/dt when x = pi/10, given dx/dt = -4 when x = pi/10.
Enter the exact answer.
dy/dt = _______
