calculus final
A cylindrical can is to be designed to hold water with a volume of 10*Pi cubic meters. What is the minimum total cost of construction of the cylinder given that the material for (i) the top costs $ 20 per square meters, (ii) the bottom costs $ 10 per square meters and (iii) the side costs $15 per square meters?
$827
Add the following two vectors. Vector C (10, 2) and Vector D (3, -2)
(13,0)
If Newton's Method is used to estimate the real root of the equation x^4 - 2x^3 + 3 = 0, then a first approximation of x = -3 would yield what value as a second approximation of x?
-2.15
With an initial guess of 1 (x = 1), what is the next approximation when using Newton's Method for this function f(x) = (1/x) - 5?
-3 Newton's equation: xn + 1 = xn - [f( xn ) / f '( xn)] Calculating the components: xn = 1 f(1) = 1/1 - 5 = -4 f ' (x) = -1/( x^2) f ' (1) = -1/1 = -1 Substituting: xn + 1 = 1 - [(-4 / -1)] = 1 - 4 = -3
Consider the function f(x) = x^4 - 5x^2 + 2x + 7. Using Newton's method for x = 0, what is your next guess?
-7/2
Estimate the integral of the function y = -x^2 + 2x. Use left Riemann Sum with x from 1 to 2 and 4 equal slices.
.782
For the function f(x) = x^2 - cos(x), use Newton's method to approximate the solution between x =0.5 and x =1 to 2 decimal places. Use radians when calculating.
.82
The limit of f(x) = sin(x) as x approaches π is _____.
0
Using Rolle's Theorem, there is a curve between the points f(3) and f(5) of a certain function, f(3) = f(5) and the curve peaks at f(4). Which of the following is the slope of the derivative of f(4)?
0
What is the limit of f(x) = sin(x) as x approaches 0?
0
What is the limit of the graph of f(x)=1/x as x approaches infinity ?
0
What is the slope of a line drawn tangent to the graph of the function f(x)=sin(x) when x = ½ pi ?
0
What is the y-value of the derivative of a function at its maximum and minimum points if the maximum is at x = 4 and the minimum is at x = 7?
0
Given dy/dx = x^3 - x Find y given y(0) = -8
0.25x^4 - 0.5x^2 - 8
When using Riemann sums, which of the following interval lengths will produce the most accurate answer?
1
Given the following expression, what is the implied interval on x? |x - a| < 1
1 + a < x < 1 + a.
Give an approximate solution, to 3 decimal places, for the function y = 2x^5 - 9. Use Newton's method and use 1.20 as starting point.
1.351
An abc right triangle has two sides measuring 6 and 8 inches respectively. What is the length of the third side?
10
If the limit of f(x) is 5 and the limit of g(x) is 2, what is the limit of f(x)*g(x)?
10
What is the area under the line y = 2x + 5, between x = -1 and x = 1?
10
A square pyramid has a height of 2 feet. If one side of the base measures 4 feet, calculate the volume of the pyramid.
10.7 cubic ft
If a prism has a volume of 33m 3, what is the volume of a pyramid that has the same base and height of that prism?
11m^3 (1/3)(33m^3)
The population of the land of Arnia is 12,000. If it has an annual growth rate of 8%, what will the population be in 4 years?
16526
A laptop monitor is 8 inches high and 15 inches wide. Calculate the diagonal length of its screen.
17
Using the midpoint Riemann Sum method, what is the area under the curve y = 2x^2 from x = 0 to x = 3? Use delta x= 1.
17.5
The two shorter sides of a right triangle measure 12 and 14 feet. What is the measurement of the hypotenuse?
18.43ft
A ladder is 20 feet long. The ladder is placed flat on the ground and is resting against a vertical wall. As a precautionary measure, the bottom of the ladder must not be closer than 3.2 feet from the bottom of the wall. How far up the wall, to 1 decimal place, can the ladder reach?
19.7
Assuming the limit of f(x) as x approaches C is infinity/infinity and the limit of f'(x) as x approaches C is 2, what is the limit of f(x)?
2
What is the area between the lines y = 2x and y = x, between x = 0 and x = 2?
2
What is the area under the line defined by y = 2 - x between x = -5 and x = 5?
20
Calculate the minimum perimeter of a rectangle that has an area of 40 square meters.
25.3
The limit of f(x) = sqrt x as x approaches 9 is _____.
3
5-3y / 3x - 2. The following is a result of implicit differentiation of dy/dx of which of the following functions?
3xy = 5x + 2y
If the limit of f(x) is 10 and the limit of g(x) is 6, what is the limit of f(x) - g(x)?
4
Which of the following can best be used as an initial guess for Newton's Method for the function y = x^5 - 17x^3?
4
f(x) = 4x - x^3. What is the area between f(x) and the x-axis from x=0 to x=2?
4
What is the area between the lines y = 10, y = 4, x = 0, x = 1?
6
When there is a y value of 7 of the derivative at x=3, which of the following would be the function of the slope?
7 ay x=3
Eric bought 1,000 feet of fencing in order to build a pen with four parallel partitions. What dimensions of the pen will lead to the maximum area of the pen?
83.3ft by 250ft
Which of the following measurements produces a right triangle?
All
If we want to find the surface area of the inside of a swimming pool with the dimensions 2 m deep x 10 m wide x 30 m long, then why is this surface area different from a rectangular prism with the dimensions 2 m deep x 10 m wide x 30 m long?
Because we're not finding the area of the surface of the water, just the rectangles that form the inside of the pool.
You can enclose a rectangular yard with a fence. The length of the longest side of the fence is going to be 2 times the length of the shortest side of the fence. Write the equation for the perimeter of the fence.
P = 6x short side= x long side= 2x P= x+x+2x+2x=6x
Differentiation of which of the following would result in -cos(x)?
The third derivation of sin(x).
A vector is best described as which of the following?
a scalar quantity plus direction
Asymptotes can be: I. Horizontal II. Oblique III. Vertical
all
Which of the following equations represent(s) a line that goes through the points (2,3) and (1,1)? I) y = 2(x - 2) + 3 II) y = 2x - 1 III) y = 2(x - 1) + 1
all
What type of discontinuity is in the graph of f(x) = 1/(x+2)?
asymptotic
Which of the following is a type of function discontinuity?
asymptotic
y = cosx Given y below and the intermediate value theorem, how many times will y = 0 between x = 0 and x = π? (Hint: Work in radians since one of the parameters for x is π.)
at least one time
If you were to graph the altitude of a jet plane in flight as a function of time it would be _____.
continuous
y = 3x - x^3/3 + C is the solution to which of the following?
dy + x^2 dx = 3 dx
y = x^2/2 + 3x + C is the solution to which one of the following differential equations?
dy - 3dx = xdx
The following is a solution to which of these differential equations? y=2+Ce^kt
dy/dt= k(y-2)
What is the function if the integral is e^x + C?
e^x
When driving over mountains, the grade of the road tells drivers how steep the climb is going up or down the mountain. The grade is the rate of change of _____.
elevation as a function of distance
Which of the following best describes when there is a minimum or maximum value?
extrema
Find the second derivative of f(t) shown below. f(t) = sin(t) + t^2
f''(t) = -sin(t) + 2
Find the second derivative of f(x) shown below: f(x) = x^4
f''(x) = 12x^2
f(x) = 45x Calculate the derivative of f(x), and evaluate it at x=2.
f'(2)45
According to Rolle's Theorem, when f(0) and f(3) results in the slope of the derivative being 0, which of the following best fits the situation?
f(0)= 3 and f(3) = 3
f(x)=18x - 24 is the result of the differentiation of which of the following?
f(x) = (3x-4)^2
Which of the following is the function if the integral is 50?
f(x) = 10 between x = 2 and x = 7.
Which of the following is the function if the integral is 8
f(x) = 4x between x = 0 and x = 2
Cos(x) is the result of the differentiation of which of the following?
f(x) = sinx
Which of the following functions has a graph in the shape of a V?
f(x) = |x|
4 cos(2x) * sin(2x) is the result of the differentiation of which of the following?
f(x)= sin^2(2x)
When the equation is f'(x) = 21, which of the following is the graph derivative of the function?
f(x)=21
f'(x) = 4 cos (4x - 2) is the result of the differentiation of which of the following?
f(x)=sin (4x - 2)
Which of the following is the function if the integral is 8?
fx = 4x between x=0 and x=2
2x + 3 is the result of the differentiation of which of the following?
fx = x^2 + 3x + 1
6x + 8 is the result of the second differentiation of which of the following?
fx = x^3 + 4x^2
If the limit of f(x) is 1 as x approaches 1 and the limit of g(x) is infinity as x approaches 1, what is the limit of (f(x) + g(x)) as x approaches 1?
infinity
Which of the following statements is true for the function f(x)=1 for x<0 and 2x for x>0 ?
it has a jump discontinuity at x=0
If the hypotenuse of a right triangle equals 1, the sine of an angle equals _____.
its opposite side
If John is going west and Robert is going east. They started at the same point and at the same time and an hour later they are 130 miles apart. Which of the following could be the speeds they are going?
john 60mph, robert 70mph
Which type of discontinuity is shown in the function f(x) = round(x)?
jump
The derivative will have a y value of zero for which of the following instances?
max and mins of the graph
The following is an example of the _____ rule for differentiating products of functions. d/dx f(x)g(x) = f(x)g'(x) + f'(x)g(x)
product
Acceleration is the measure of how velocity changes over time. Acceleration is an example of _____.
rate of change
A bullet moves with an acceleration defined by a(t) = 2t. If velocity is constant, v = 1, and the shooter takes a step equaling 2 feet, making s = 2, what is the equation for the position as a function of time?
s(t) = 1/3(t^3) + t + 2
What would the cosine of an angle be equal to if the hypotenuse of a right triangle is 1?
the adjacent side
In the formula for calculating an arithmetic series, what does the following term represent? "a1"
the first term in the series
The derivative of f(x) is equal to 4. What does this mean?
the instantaneous rate of change is equal to 4
The limit of f(x) / limit of g(x) is the same as _____.
the limit of (fx/gx)
If you have a function, x as a function of time and x(t) is your position, then:
the second derivation is the acceleration
One of the steps involved in optimizing systems involves writing an equation. What is a possible alternate to writing the equation?
there is no alternative
If a family going on vacation starts out from its driveway, then travels 60 miles in the first hour of driving, which of the following statements must be true?
they must have exceeded 60 mph at some point during the hour
The Mean Value Theorem is used to determine if someone was speeding in between _____.
two points
If a continuous function starts below the x axis at x = -3 and is above the x axis at x = 4, the Intermediate Value Theorem states the function could have a solution at _____.
x = 1
If I am currently at x sub 3 and I need to keep going to find a better answer, what will my next guess be?
x sub 4 = x sub 3 - y(x sub 3)/f'(x sub 3)
For the function f(x) = x for x > 4 and f(x) = 4 - x for x < 4, which of these has a continuous range?
x>4
Which of the following is the best possible answer if the integral is 9?
x^2 between x = 0 and x = 3?
Expand f(x) = (x+3)^2.
x^2+6x+9
Which of these keys would you use to find 6 to the 8th power?
x^y
Which of the following answers best fits when the area between the lines is 8?
y = 4 and y = 2 between x = 0 and x = 4?
Which of the following answers best fits when the area between the lines is 12?
y = 4x - 10 between x = 3 and x = 5.
What is the equation of the line passing through the point (3,2) with a slope of 5?
y = 5x-13
When x = -2 and x = 1, the function has a height of -2. Which of the following formulas best depicts this scenario?
y=x^3-3x
sec(x) + c = _____
{sec x tan x dx
Given a watermelon with a length of 12 inches and a diameter of 6 inches, which of the following functions could be used to find the volume using the disk method for volumes of revolution?
π(3)^2dx
