Application of derivatives Questions and Answers

Let h' (u)=6/u Find the particular solution to the above differential equation that satisfies the following initial condition. h(e^8) =-2 h(u) =

Let h''(v) = 0 Find the particular solution to the above differential equation that satisfies the following initial conditions. h'(8) = - 4 h(9) = -6 h(v)=

Let dh/dx = 3x - 3 Find the particular solution to the above differential equation that satisfies the following initial condition. • h(8) = -8 h(x)=

Consider the following function. f(x) = x³ - 3x2 + 3x Find the derivative of the function. f'(x) = Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. This function is strictly monotonic and therefore has an inverse. This function is not strictly monotonic and therefore does not have an inverse. It can not be determined whether or not this function is strictly monotonic.

Find the general solution of the differential equation below and check the result by differentiation dp/dz=-16z^7 P(z)=-2z^8+1 P(z)=-2z^8 P(z)=-112z^6 P(z) = -2z^8+ C P(z)=-112z^6 +C

Find f. f"(x) = −2 + 12x - 12x², f(0) = 8, f'(0) = 14

Find the vertex of the graph of the function. Do not enter any spaces in the answer. f(x) = (x+80)² - 21

Find the point on the line y = 4x + 2 that is closest to the origin. (x, y) =

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle, labeled x.) If the perimeter of the window is 8 feet, find the exact value of x (in ft) so that the greatest possible amount of light is admitted.

Find the (a) period, (b) phase shift (if any), and (c) range of the following function. y=-3-tan (x + π 2)

Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points. f(x) = -x6 + 30x5 - 30x +3 For what interval(s) of x is the graph of f concave upward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The graph is never concave upward. For what interval(s) of x is the graph of f concave downward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (Type an exact answer. Use a comma to separate answers as needed.) B. There are no inflection points. (Type your answer in interval notation. Type an exact answer. Use Determine the x coordinates of any inflection points of the graph of f(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. separate answers needed.) A. x= B. The graph is never concave downward.

Let g(x) = 6x²-18x+8, f₁(x) = |g(x)|, f₂(x) = |f₁(x) - P₁|, f3(x) = |f₂(x) - P₂| and if P₁ = 7 then the range of P₂ such that f3(x) has exactly 10 points of non- differentiability is, (A). (1,5,7) (B). [2,5,8] (C). [2,9] (D). (1,8)

Find two numbers whose sum is 25 and whose product is a maximum.

Find two positive numbers whose product is 22 and whose sum is a minimum. The two numbers are (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

Find two numbers whose sum is 24 and whose product is a maximum. The two numbers are (Simplify your answer. Use a comma to separate answers as needed.)

Which slope field is associated with the differential equation given below? dy/dx= 1-y

Gonzalez Manufacturing borrowed $33000. Part of the money was borrowed at 10%, part at 12%, and part at 14%. The annual interest was $3980, and the total amount borrowed at 10% and 12% was twice the amount borrowed at 14%. Use Gaussian elimination or Gauss-Jordan elimination to find the amount borrowed at each rate. How much money was borrowed at 10%? How much money was borrowed at 12%? How much money was borrowed at 14%?

Find the x and y coordinates of all inflection points. f(x) = x³ +27x² What is/are the inflection point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The inflection point(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no inflection points.

Find the x and y coordinates of all inflection points. f(x)=x³ +21x² What is/are the inflection point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The inflection point(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no inflection points.

Of all numbers whose difference is 12, find the two that have the minimum product. What are the two numbers? (Use a comma to separate answers as needed.)

Find two positive numbers whose product is 91 and whose sum is a minimum. The two numbers are (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

Find two numbers whose sum is 22 and whose product is a maximum. The two numbers are (Simplify your answer. Use a comma to separate answers as needed.)

Of all numbers whose difference is 16, find the two that have the minimum product. What are the two numbers? (Use a comma to separate answers as needed.)

Inspect the graph of the function to determine whether it is concave up, concave down or neither, on the given interval. A cube function, m(x)=x^3, on (-∞,0) On the interval (-∞o,0), the function m(x) = x is concave up. neither concave up or concave down. concave down.

Find two numbers whose sum is 9 and whose product is a maximum. The two numbers are (Simplify your answer. Use a comma to separate answers as needed.)

Suppose f(x) = ln (3)3x. Determine and graph four distinct antiderivatives of f.

Number 2.) 8, 21, 46, 89, 156, ... General Rule = ? 23rd term = ? 32nd term = ?

First use appropriate properties of logarithms to rewrite f(x), and then find f'(x). f(x) = 4 In(2/x)

Simplify the expression as much as possible. tan(x) cos(x) =

If possible, find a solution to tan(4θ + 1) = -4. If no solution exists, enter NONE. radians help (angles) θ

Consider the following curve. y =X+3/X-4 Find the slope m of the tangent line at the point (3, -6). m = y = Find an equation of the tangent line to the curve at the point (3, -6).

How many years will it take $9,000 to grow to $14,200 if it is invested at 3.50% compounded continuously?

Consider h(v) = 13v / (v²+18) Determine the intervals on which h is decreasing. h is decreasing on: h is decreasing nowhere. Determine the intervals on which h is increasing. h is increasing on: h is increasing nowhere. Determine the value and location of any local minimum of h. Enter the solution in (v, h(v)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. h has a local minimum at: h has no local minimum. Determine the value and location of any local maximum of h. Enter the solution in (v, h(v)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. h has a local maximum at: h has no local maximum.

Let g(x) = x³ +24x Determine the absolute extrema of g on [1, 5]. If multiple such values exist, enter the solutions using a comma-separated list. The absolute minimum of g is and it occurs at x = The absolute maximum of g is and it occurs at x =

At 2:00 p.m. a car's speedometer reads 30 mi/h. At 2:10 p.m. it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h². Let v(t) be the velocity of the car t hours after 2:00 p.m. Then [v(1/6) - v(0)] / (1/6 - 0) = By the Mean Value Theorem, there is a number c such that 0 < c < with v'(c) = Since v'(t) is the acceleration at time t, the acceleration c hours after 2:00 p.m. is exactly 120 mi/h².

Find the demand function p, given the marginal-revenue function dr/dq below. dr/dq=299-8q-0.3q² Find y subject to the given conditions. y''' = 12x, y''(0) = 16, y'(0) = 6, and y(0) = 7 y(x) = (Simplify your answer. Do not factor.)

First use the appropriate properties of logarithms to rewrite f(x), and then find f'(x). f(x) = 13x + In 13x Rewrite f(x) using properties of logarithms. f(x) = (Do not simplify.) Find f'(x). f'(x) =

Find two numbers whose sum is 10 and whose product is a maximum. The two numbers are

A couple who borrow $80,000 for 30 years at 6%, compounded monthly, must make monthly payments of $479.64. (a) Find their unpaid balance after 1 year. (Round your answers to the nearest cent.) (b) During that first year, how much do they pay towards the principle? (Round your answer to the nearest cent.) During that first year, what are their total payments? (Round your answer to the nearest cent.)

Consider the following function. f(x) = 16 - x^(2/3) Find f(-64) and f(64). Find all values c in (-64, 64) such that f'(c) = 0. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Based off of this information, what conclusions can be made about Rolle's Theorem? This contradicts Rolle's Theorem, since f is differentiable, f(-64) = f(64), and f'(c) = 0 exists, but c is not in (-64, 64). This does not contradict Rolle's Theorem, since f(0) = 0, and 0 is in the interval (-64, 64). This contradicts Rolle's Theorem, since f(-64) = f(64), there should exist a number c in (-64, 64) such that f'(c) = 0. This does not contradict Rolle's Theorem, since f'(0) does not exist, and so f is not differentiable on (-64, 64). Nothing can be concluded.

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = 3x² - 4x + 2, [0, 2] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable on R. No, F is not continuous on [0, 2]. No, f is continuous on [0, 2] but not differentiable on (0, 2). There is not enough information to verify if this function satisfies the Mean Value Theorem. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE).

The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 190 cm²?

D(x) is the price, in dollars per unit, that consumers are willing to pay for x units of an item, and S(x) is the price, in dollars per unit, that producers are willing to accept for x units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point D(x)=-3x/10+15, S(x) = x/5+5 (a) Find the equilibrium point.

A Company operates 12 oil wells in a designated area, Each pump, on average, extracts 156 barrels of oil daily. Company can but well reducep every added output of each of the wells by 6 barrels. How many wells should the Company add in order to maximize daily production ?

Solve for the variable P to two decimal places. 63,300 Pe ^0.072(9) P = (Round to two decimal places as needed.)

An investor bought stock for $20,000. Eight years later, the stock was sold for $40,000. If interest is compounded continuously, what annual nominal rate of interest did the original $20,000 investment earn?

A company produces two types of solar panels per year: x thousand of type A and y thousand of type B. The revenue and cost equations, in millions of dollars, for the year are given as follows. R(x,y) = 5x + 6y C(x,y)=x²-3xy + 7y² +11x-41y-4 Determine how many of each type of solar panel should be produced per year to maximize profit.

Of all points (x,y,z) that satisfy x + 3y + 2z = 27, find the one that minimizes (x-1)² + (y-1)²+(z-1)². The given function has a minimum at which satisfies x + 3y + 2z = 27. (Simplify your answer.)

Mari needs to design an open-top box with a square base. Due to the nature of the project, the box must have a volume of 48,668 cubic inches. Determine the dimensions of the box that requires the least amount of materials to construct. Each side of the base of the box has of length ____ inches. The box has a height of ____ inches.

A cylindrical container must hold 92 cubic centimeters of fluid. Determine the radius the cylinder such that the construction materials necessary are as least as possible.