# Application of derivatives Questions and Answers

Calculus

Application of derivativesFind the points on the cardioid r = sin(2θ) where the tangent line is horizontal. Draw.

Calculus

Application of derivativesWhich integral correctly computes the length of the curve y=In(cos(x)) between x=0 and x=π/3 ?
a) π/3
∫₀tan(x) dx
b) π/3
∫₀2πln(cos(x))√(1+tan²(x)) dx
c) π/3
∫₀sec(x) dx
d)π/3
∫₀√(1 + (In(cos(x)))² )dx

Calculus

Application of derivativesJina is driving on a highway. She uses one gallon of gas every 20 miles she drives. The distance, D (in miles), she travels on g gallons of gas is given by the following function.
D(g) = 20g
How far does Jina drive if she uses 4 gallons of gas?
________ miles

Calculus

Application of derivativesSuppose that the revenue realized on the sale of company's product can be modeled by the function R(x) = 500x -0.4x² is the number of units sold and R(x) is given in dollars. (Revenue is the income a company receives from selling its goods at services. Compute the average change in revenue, per additional item that was sold for the first 50 items.

Calculus

Application of derivativesKings Dominon's opening day had 1350 people visit the park. Children under 12 paid $40, Adults under 65 paid $65 and seniors over 65 paid $50. The park had combined ticket sales of $75500. If each child was accompanied by 2 adults under the age of 65, how many of each age group were in the park?

Calculus

Application of derivativesFind the area of the sector of a circle with radius 15 feet formed by a central angle of 35°.
_______ square feet
Round your answer to two decimal places.

Calculus

Application of derivativesBelow is the function f(x) = 11 - 1/2x², together with a rectangle with one side on the x-axis, and two corners on the curve. The bottom left corner is (x, 0).
What is the area of the rectangle if x = 2.3? [Give your answer as a decimal number.]

Calculus

Application of derivativesSolve the equation for θ, where 0 ≤ θ ≤ 2. (Enter your answers as a comma-separated list.)
4 cos² (θ) = 3
θ = ____

Calculus

Application of derivativesDetermine the minimum of
f(x)= (10x³ + 3x² + x + 5)²
starting at x = 3 and using a step size △ = 5.0.
Using region elimination: expanding pattern bounding plus six steps of golden section.

Calculus

Application of derivativesIn your own words, define the following.
(a) Secant line
(b). Average rate of change
(c). Tangent line
(d). Instantaneous rate of change

Calculus

Application of derivativesGiven that sin(a) = 1/3 and cos(b) = -2/6 with a and b both in the interval [ π/2, π], find the exact values of sin(a + b) and cos(a − b).

Calculus

Application of derivatives• The volume of a cube is increasing at a rate of 10cm3/min. How fast is the surface area increasing when the length of an edge (side) is 30 cm (related rates)
•The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm.
a) Use differentials to estimate the maximum error in computing the volume of the cube.
b) What is the relative error? What is the percent of error?

Calculus

Application of derivativesGiven the functions f(x) = x² + 1 and g(x) = x + 1. Compute the net and the total (or absolute) area between the curves over the interval [0,2].

Calculus

Application of derivatives6. For the function y = 30 cos[2(x + 15°)] - 60, the phase shift is
A 30° to the right
B 15° to the left
C 15° to the right
D 60° to the right

Calculus

Application of derivativesSuppose is an angle such that cos(θ)= -1/7
If 2π < 0 < 3π,
then cos(θ/2)=___
and sin(θ/2)=___

Calculus

Application of derivativesFind the θ value within [0, 2π) for function f(θ) = 7 cos(7θ) to reach its minimum value.
θ =

Calculus

Application of derivativesMatch each side as hypotenuse, opposite, or adjacent of angle A.
Adjacent a. BC
Opposite b. AB
Hypotenuse C. AC

Calculus

Application of derivativesDetermine the center and radius of a circle given that its equation is
(x - 1)² + (y- 8)² = 49
Center:
Radius:

Calculus

Application of derivativesFind the θ value within [0, 2π) for function f(θ) = 9 sin(10 θ) to reach its maximum value.
θ =?

Calculus

Application of derivativesThe cost, in dollars, to produce a vats of ice cream is C(x) = 9x + 14. When selling them to ice cream shops, the price-demand function, in dollars per vat, is p(x) = 87 - 3x
Find the profit function.
P(x) =
How many vats of ice cream need to be sold maximize the profit.
Find the maximum profit.
Find the price to charge per vat to maximize profit.

Calculus

Application of derivativesFind the θ value within [0, 2π) for function f(θ) = 9 sin(10θ) to reach its maximum value.
θ =

Calculus

Application of derivativesIn Exercises 57-62, write the slope-intercept equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.
Point
57. (2, 1)
62. (-4,1)
Line
4x - 2y = 3
x+y=7
y +2=0

Calculus

Application of derivativesA true-false test consists of 16 questions.
a) In how many ways can the test be completed, selecting true or false for each question?
b) What is the probability that a test is randomly answered perfectly?
a) In how many ways can the test be completed, selecting true or false for each question?
The test can be completed in ways.
(Type a whole number.)
b) What is the probability that a test is randomly answered perfectly?
The probability is
(Type an integer or a simplified fraction.)

Calculus

Application of derivativesSuppose E and F are events such that n(E) = 12, n(F) = 16, and n(ENF) = 3.
a) Find P(EIF).
b) Find P(FIE).
a) P(EIF) =
b) P(FIE) =
(Type an integer or a simplified fraction.)
(Type an integer or a simplified fraction.)

Calculus

Application of derivativesComplete the parametric equations for the line where the planes -2x-4y-4z=-8 and -15x-y-10z=-29 intersect.
x(t) = 36t:
y(t) = _______________________
z(t) = _______________________

Calculus

Application of derivativesThe plane containing the point (0,-3,1) and parallel to the plane
-5x-7y-8z=7 has equation Ax+By+Cz= D.
If A=-5 find the following:
B = -7
C=-8
D=15

Calculus

Application of derivativesIndicate whether the table specifies a function.
Domain Range
- 14 4
-11 -11
17 2
-3 15
The table is a function.
The table is not a function.

Calculus

Application of derivatives1)Find the slope of the secant line for the function f(x) = (-1/2)(x²-3x - 10) passing through (1, 6) and (4,3).
2). Find the average rate of change off(x) = x² over the interval [-1,1].

Calculus

Application of derivativesWrite the first three terms in the binomial expansion, expressing the result in simplified form.
(3x+2y)7

Calculus

Application of derivativesFind the center, foci, and vertices of the ellipse. Graph the equation.
4x²+36y²-24x+288y +468 = 0

Calculus

Application of derivativesIn an experiment on social interaction, 6 people will sit in 6 seats in a row. In how many ways can this be done?
There are ___ ways.

Calculus

Application of derivativesCharles and Bernice ("Ray") Eames were American designers who made major contributions to modern architecture and furniture design. Suppose that a manufacturer wants to make an Eames elliptical coffee table 78 in long and 62 in wide out of an 8 ft by 4 ft piece of birch plywood. If the center of a piece of plywood is positioned at (0, 0), determine the distance from the center at which the foci should be located to draw the ellipse. Express numbers in exact, simplest form.

Calculus

Application of derivativesOf the entering class at a college, 69% attended public high school, 25% attended private high school, and 6% were home schooled. Of those who attended public high school, 19% made the Dean's list, 18% of those who attended private high school made the Dean's list, and 17% of those who were home schooled made the Dean's list. A student is randomly chosen.
a) Find the probability that the student made the Dean's list.
b) Find the probability that the student came from a public high school, given that the student made the Dean's list.
c) Find the probability that the student was not home schooled, given that the student did not make the Dean's list.
a) The probability that the student made the Dean's list is ___
(Do not round until the final answer: Then round to four decimal places as needed.)
b) The probability that the student came from a public high school, given that the student made the Dean's list, is ___
(Do not round until the final answer. Then round to four decimal places as needed.)
c) The probability that the student was not home schooled, given that the student did not make the Dean's list, is ___
(Do not round until the final answer. Then round to four decimal places as needed.)
BICEERD

Calculus

Application of derivativesFor (y-6)²/25-(x+5)²/16=1
(a) Identify the center.
(b) Identify the vertices.
(c) Identify the foci.
(d) Write equations for the asymptotes.
(e) Graph the hyperbola.
Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.
Part: 1 / 5
(a) The center is ( , ).
Part 2 of 5
(b) The vertices are ( , ) and ( , ).

Calculus

Application of derivativesKevin wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Kevin has 850 feet of fencing, what dimensions would maximize the area of the pen?
a) Let ω be the length of the pen perpendicular to the barn. Write an equation to model the area of the pen in terms of ω.
Area = ______________
b) What width ω would maximize the area?
ω = ______________

Calculus

Application of derivativesA person on earth is observing the moon, which is 238,860 miles away. The moon has a diameter of 2160 miles. What is the angle in degrees spanned by the moon in the eye of the beholder? Note, consider the diameter of the moon as an arc length for this question. Give your solution rounded to 3 places after the decimal.

Calculus

Application of derivativesA telephone company determines that the duration t, in minutes, of a phone call is an exponentially distributed random variable with a probability density function f(t) = 3 e 3t, 0≤t<∞o. Find the probability that a phone call will last no more than 1.5 min. The probability is __

Calculus

Application of derivativesSolve ΔABC subject to the given conditions, if possible. Round the measures of the angles (in degrees) to 1 decimal place if necessary.
a=2.8, b=4, c = 14.2
The triangle with these conditions does not exist.
The triangle with these conditions does exist.
A≈_
B≈_
C≈_

Calculus

Application of derivativesFind the maximum value of f(x, y) = 4x² - 6y² + 15 on the disk
x² + y² ≤ 1

Calculus

Application of derivativesA hot brick is removed from a kiln at 225° C above room temperature. Over time, the brick cools off. After 2 hours have elapsed, the brick is 30° C above room temperature. Let t be the time in hours since the brick was removed from the kiln. Let y = H(t) be the difference between the brick's and the room's temperature at time t: Assume that H(t) is an exponential function.
Interpret the physical meaning of the horizontal asymptote of H(t).

Calculus

Application of derivativesConsider the following curve.
F(X)=X-6/X2
Find the local minimum and maximum values of f. (Enter your answers as comma-separated lists.
local minimum values____
local maximum values_____

Calculus

Application of derivativesIf a tank holds 3500 gallons of water, which drains from the bottom of the tank in 50 minutes, then Toricelli's Law gives the volume V of water remaining in the tank after t minutes as - 3500 (1-¹² 50 V = (b) 10 min (c) 20 m Find the rate (in gal/min) at which water is draining from the tank after the following amounts of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.) (a) 5 min (d) 50 min gal/min t = gal/min gal/min 2 gal/min 0 ≤ t ≤ 50. At what time (in min) is the water flowing out the fastest? t = min At what time (in min) is the water flowing out the slowest? min

Calculus

Application of derivativesLet h(x)=-7+ 5/x
Determine the average value, h(c), of h over [6, 8].
h(c) =
Determine the value(s) of c in [6, 8] guaranteed by the Mean Value Theorem.
c=

Calculus

Application of derivativesA population is growing according to the exponential model given by P (t) = 500e0.25t, where t is measured in years. How many years will it take for the population to reach 1000? (Round your answer to one decimal place.)

Calculus

Application of derivativesThe population of trout in a pond is given by P = 0.4P(1 - P 10,000) -f, where f trout are caught per year.
Consider the case where a certain number of fish are added to the pond, or f = -300. What are the nonnegative equilibria and their stabilities? (Round your answer to two decimal places.)

Calculus

Application of derivatives4. Solve the initial-value problem
y"(t)- y'(t)- 6y(t) = 0, y(0) = 2, and y'(0) = 1.
[Hint: First determine the solution by computing the roots of the associated quadratic equation, then determine the values of the constants using the given initial conditions]

Calculus

Application of derivativesFind the general power series solution centered at Xo= 0 for the differential equation
y' - 2xy = 0

Calculus

Application of derivativesConsider the first-order equation with initial condition
dy/dt=-y+t² y(0) = 1
Use a value of Ax = h = 1.0 to find y(1) by the third-order Runge-Kutta (RK) method, showing all equations and work.