Limits & Continuity Questions and Answers

Find the intervals on which f(x) is increasing, the intervals on which f(x) is decreasing, and the local extrema. f(x) = (x+3) ex Compute the derivative of f(x). f'(x) = Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is increasing on (Type your answer using interval notation. Type integers or simplified fractions. Use a comma to separate answers as needed.) B. The function is never increasing. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is decreasing on (Type your answer using interval notation. Type integers or simplified fractions. Use a comma to separate answers as needed.) B. The function is never decreasing.

Find the consumer surplus at the equilibrium point. Round to the nearest cent. D(x)=(x-3)²; x =3/2 A. $7.28 B. $3.25 C. $4.33 D. $4.50

Find the present value of $11,700 due 7 yr later at 12.5% compounded continuously. Round the answer to the nearest cent. A. $3,707.29 B. $4,877.29 C. $28,066.82 D. $26,896.82

Find the equilibrium point. D(x)=9-6x, S(x) = 4 + 6x A. (5/12, $19) B. (-5/12, $1.5) C. (5/12, $6.5) D. (5/2, $19)

Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2 cos(t) + sin(2t), [0,π/2]

1000 dollars grows to 1 million dollars after 60 years in a bank. If interest is compounded continuously, what is the rate of interest per year? a 1.83% b. 11.51% c. 28.13% d. 3.84% e 0.12%

Bob and Doris Mackenzie have a new grandchild, Brenda. They want to create a trust fund for her that will yield $250,000 on her 24th birthday. a) What lump sum would they have to deposit now at 5.6%, compounded continuously, to achieve $250,000? b) Bob and Doris decide instead to invest a constant money stream of R(t) dollars per year. Find R(t) such that the accumulated future value of the continuous money stream is $250,000, assuming an interest rate of 5.6%, compounded continuously.

At age 35, Clarence earns his MBA and accepts a position as a vice president of an asphalt company. Assume that he will retire at the age of 65, having received an annual salary of $90,000, and that the interest rate is 5%, compounded continuously. a) What is the accumulated present value of his position? b) What is the accumulated future value of his position?

In 17 years, Maggie Oaks is to receive $100,000 under the terms of a trust established by her grandparents. Assuming an interest rate of 5.4%, compounded continuously, what is the present value of Maggie's legacy?

In 2004 (t = 0), the world consumption of a natural resource was approximately 18 trillion cubic feet and was growing exponentially at about 10% per year. If the demand continues to grow at this rate, how many cubic feet of this natural resource will the world use from 2004 to 2007?

Find the accumulated present value of a continuous stream of income at rate R(t) = $132,000, for time T = 17 years and interest rate k= 3%, compounded continuously.

Find the present value Po of the amount P = $200,000 due t = 6 years in the future and invested at interest rate k = 7%, compounded continuously.

Find the future value P of the amount Po invested for time period t at interest rate k, compounded continuously. Po = $200,000, t = 7 years, k = 4.8%

Solve for r to two decimal places. 3= e ^3r

Solving A = Pe^rt for P, we obtain P = Ae^¯rt which is the present value of the amount A due in t years if money earns interest at an annual nominal rate r compounded continuously. For the function P = 11,000e-0.04t, in how many years will the $11,000 be due in order for its present value to be $9,000?

The price-earnings ratio of a stock is given by R(PE)=P/E where P is the price of the stock and E is the earnings per share. Recently, the price per share of a certain company was $30.28 and the earnings per share were $1.18. Find the price-earnings ratio

Determine the domain of the function of two variables. g(x,y)= 8 / 2y-7x^2

Of all the numbers whose sum is 32, find the two that have the maximum product.

Determine two positive values such that the sum of the two numbers is 216 and the product of the first number and the square of the second number is a maximum. Enter the solutions using a comma-separated list. Determine the maximum value.

Find a positive number such that the sum of its reciprocal and 6,912 times its square is as small as possible.

Graph all three functions, sin(x), 2 sin(x), sin(2x), on one axis. Make sure to label your functions.

Determine two positive values such that the product of the two values is 10658 and the sum of the first number and two times the second number is a minimum. Enter the solutions using a comma-separated list. Determine the minimum value of the sum.

A firm has the marginal-demand function D'(x) = - 1800x/√(25-x2), where D(x) is the number of units sold at x dollars per unit. Find the demand function given that D = 17,000 when x = $4 per unit.

Let f(x) = 8 - 13x² - 2x5 + 3x6. Find the following: Degree of the f(x) = Leading coefficient End behavior: (Note: type "infty" for ∞ and "-infty" for -∞ ) As x →-∞, f (x) → As x →∞, f (x) → Maximum number of intercepts: Maximum number of turning points:

Find the missing factor, A. x (A) = -14x²

Use long division to find the quotient and remainder. (2x² + 9x - 12) ÷ (x + 5) What is the quotient? What is the remainder?

Find the extreme values of f on the region described by the inequality. (If an answer does not exist, enter DNE.) f(x, y) = e^-xy, x² + 4y² ≤ 1 maximum minimum

If the length of the diagonal of a rectangular box must be L, what is the largest possible volume? cubic units

Find the x-values (if any) at which f(x)=x/x^2 -2x is not continuous. f(x) is not continuous at x=0 and f(x) has a removable discontinuity at x=0. f(x) is not continuous at x=0, 2 and both the discontinuities are nonremovable. f(x) is not continuous at x=2 and f(x) has a removable discontinuity at x=2. f(x) is not continuous at x=0,2 and f(x) has aremovable discontinuity at x=0. f(x) is continuous for all real x.

Heather has $45.71 in her savings account. She bought six packs of markers to donate to her school. Write an expression for how much money she has in her bank account after the donation. A. 45.71-6m B. 45.71 +6m C. 45.71-6 D. 45.71 +6

Determine the leading coefficient and degree of the polynomial. (2x+5)(x-1)(3x + 2)² a. The leading coefficient is b. The degree is

Evaluate the polynomial function for x = -4 and x = -1. f(x)=x²-5x - 14 f(-4)= f(-1) = Based on the results and the Intermediate Value Theorem, which statement is correct? Because both f(-4) and f(-1) are positive, f has no real zeros between x = -4 and x = -1. Because f(-4) is positive and f(-1) is negative, f has exactly one real zero between x = -4 and x = −1. Because f(-4) is negative and f(-1) is positive, f has at least one real zero between x = -4 and x = −1. Because f(-4) > f(-1), f has at least one real zero between x = -4 and x = -1. Because f(-4) is positive and f(-1) is negative, f has at least one real zero between x = -4 and x = -1.

Solve the following equation. 3x³ - 6x² - 4x + 8 = 0

Perform the indicated operation and simplify the results. Assume that the denominator in each problem is not 0. a) Divide 6m² + 2m by 2m b) Divide 8t⁴+ 4t³ by 4t² c) Simplify 16x⁶ + 20x³ -12x / 4x²

Solve for the variable P to two decimal places. 60,900 = Pe^0.099(12) P = (Round to two decimal places as needed.)

The range R of a projectile is related to the initial velocity v and projection angle by the equation R = v² sin 2θ/g, where g is a constant. How is dR/dt related to dv/dt if θ is constant?

The elasticity of a particular thermoplastic can be modeled approximately by the relation ε = 2.5x10⁵/T^2.3, where T is the Kelvin temperature. If the thermometer used to measure T is accurate to 1%, and if the measured temperature is 476 K, how should the elasticity be reported?

Suppose that the dollar cost of producing x radios is c(x) = 200 + 10x - 0.2x². Find the average cost per radio of producing the first 30 radios.

Given g(x)=x²-9x, find the equation of the secant line passing through (-3, g (-3)) and (5, g (5)). Write your answer in the form y=mx+b.

The one-to-one function g is defined below. g(x) = 9x/5x-6 Find g⁻¹(x), where g⁻¹ is the inverse of g. Also state the domain and range of g⁻¹ in interval notation. g⁻¹(x) = Domain of g ¹ : Range of g⁻¹ :

Write 7 sinh(x) + 4 cosh(x) in terms of e^x and e^-x.

Find all solutions of the equation. 2 sin x + 1 = 0

Find the derivative of Fx)=2x-5 using limits

Find the exact value by using a sum or difference identity. sin 165°

For eac of functions f and g below, find f(g(x)) and g (f(x)). Then, determine whether fand g are inverses of each other. Simplify your answers as much as possible. (a) f(x) = x/6 g(x) = 6x f(g(x)) = g(f(x)) = fand g are inverses of each other fand g are not inverses of each other (b) f(x) = x-5/2 g(x) = 2x + 5 f(g(x)) = g (f(x)) = 0 f and g are inverses of each other f and g are not inverses of each other

Prove that k= 1/s , tau =0 s> o are the natural equations of the logarithmic spiral given by r= 1/√2 e^θ

If you translate the graph of y = f(x) 4 units to the right, reflect it over the x-axis, stretch it vertically by a factor of 2, then translate it 5 units up, what is the equation of the new graph?

Use sigma notation to write the sum. 5/3+1 + 5/3+2 + 5/3+3 + ... + 5/3+14

Construct a variation of the Recaman sequence numerically and also with a picture and formula. This variation is that you do not increase by one, but by even numbers. Thus move up or back 2, then 4, then 6, etc.

Determine if the function y = x+1 / x^2 - 3x + 2 is continuous at the values x=0, x= 1, and x = 2. Is the function continuous at x = 0? Is the function continuous at x = 1? Is the function continuous at x = 2