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Given f(x)=(x^2+3*x+1)^5 / (x+3)^5 , identify a function u of x and an integer n not equal to 1 such that f(x)=u^n. Then compute f'(x).

Divisibility Tests and Rules and Quotient Polynomials

Let n be a positive integer a) prove that n is divisible by 5 if and only if it ends with 0,5 b) prove than n is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 c) find a similar criterion for divisibility by 7 and prove it .

Revenue, Supply and Demand Functions : Derivatives and Integrals

5. A man was sentenced to 50 years in prison when he was 20 years old. While in prison he reflected on his life and decided that he should turn his life around and do something good for his society. He then became a model prisoner and his good behavior earned him the privilege to pursue a career in law. When he became 39 years

Implicit differentiation

Consider this equation: x2 - 2xy + 4y2 = 64 A) write an expression for the slope of the curve at any point (x,y) B) Find the equation of the tangent lines to the curve at the point x = 2 C) find d2y/dx2 at (0,4)

Critical Numbers, Derivatives and Rates of Change

See the attached file. The function has one critical number. Find it. A student decided to depart from Earth after his graduation to find work on Mars. Before building a shuttle, he conducted careful calculations. A model for the velocity of the shuttle, from liftoff at t = 0 s until the solid rocket boosters were jettisone

Applications of Derivatives Word Problems and Rate of Change

A street light is at the top of a 14 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 40 ft from the base of the pole? Note: You should draw a picture of a right triangle with the vertical side representing the pole,

Interest and applications of derivatives.

A person's fortune increases at a rate to the square of they're present wealth. If the person had one million dollars a year ago and has two million today then how much will the person be worth in six months?

Directional Derivatives

Consider the function f(x,y,z) = (e^z)ln(x^2 + y^2) a) Is there a vector r such that the directional derivative of f at (1,1,0) in the direction of r equals 1? If there is, find one such vector. If not, explain why not. b) Is there a vector r such the directional derivative of f at (1,1,0) in the direction of r equals to

Functions: Linear Regression, Derivatives and Rate of Change

See the attached file. 1. A college calculus professor wanted to investigate the relationship between student's scores on the first exam and the overall course grades. A sample of the data is below. (All values are given in percents.) first exam score 54 98 73 100 88 90 77 73 81 final grade % 60 93 69 95 82 87 72 71 74

Inverse functions

Please see problems and show step by step solution in detail please. --- 7.4 Inverse functions Differentiate the problems: 1) f(x) = ln(x^2 + 10) 2) f(à?) = ln(cos à?) 3) f(x) =log2(1-3x) 4) f(x) = 5thROOT(ln x) 5) f(x)=SQRTx * (ln x) 6) f(t) = ln [(2t+1)^3 / (3t-1)^4] 7) h(x)=ln(x + SQRT(x^2-1)) 8) g(x)=ln[(


1.) compute the derivative of f(x)= arctan (x^2) 2.) compute the derivative of f(x)= ln(x^2/(2+x)) 3.) determine an equation for the line tangent to the graph of y= xe^x at the point on the graph were x=2

Application Word Problem: Continuity and Derivatives

It costs Sugarco 25 cents/lb to purchase the first 100 lb of sugar 20 cents/lb to purchase the next 100 lb and 15 cents to buy each additional pound. Let f(x) be the cost of purchasing x pounds of sugar. Is f(x) continuous at all points? Are there any points where f(x) has no derivative?


Please solve and explain. Two factories are located at the coordinates (-x,0) and (x,0), with their power supply located at (o,h). Find y such that the total amount of power line from power supply to the factories is a minimum.


A rectangular package can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume. Assume cross section is square.

Maximum area with given perimeter

A Norman window is constructed by adjoining a semicircle to the top of a rectangular window. Find the dimensions of the Norman window of maximum area if the total perimeter is 16 feet.


Use a graphing utility to graph f and g in the same window and determine which is increasing at the faster rate for "large" values of x. What can you conclude about the rate of growth of the natural logarithmic function? f(x) = ln x, g(x) = the square root of x


Find the length and width of a rectangle that has an area of 64 square feet and a minimum perimeter.

Differentiation and chemical reaction

Please solve to the specified answer and explain how to do so. In an autocatalytic chemical reaction, the product formed is a catalyst for the reaction. If Q sub zero is the amount of the original substance, and x is the amount of catalyst formed, the rate of the chemical reaction is dQ/dx = kx(qsubzero - x) For what

Graphing from the derivative

Please sketch a graph of a function f have the indicated characteristics. Please explain. (a) f(0) = f(2) = 0 f'(x) > 0 if x <1 f'(1) = 0 f'(x) < 0 if x > 1 f''(x) < 0 (b) f(0) = f(2) = 0 f'(x) < 0 if x < 1 f'(1) = 0 f'(x) > 0 if x > 1 f''(x)

S represents weekly sales of a product.

S represents weekly sales of a product. What can be said of S' and S'' for each of the following? (a) the rate of change of sales is increasing (b) sales are increasing at a slower rate (c) the rate of change of sales is constant (d) sales are steady (e) sales are declining, but at a slower rate (f) sal

Mean value theorem proof

Please explain how to prove the following. As much explanation as possible would be great Let p(x) = Ax^2 + Bx + C. Prove that for any interval [a,b], the value c guaranteed by the Mean Value Theorem is the midpoint of the interval.

Question about Differentiation proof

Please indicate if each statement is true or false and if false please explain why If the graph of a function has three x intercepts, then it must have at least two points at which its tangent line is horizontal. If f'(x) = 0 for all of x in the domain of f, then f is a constant function.

Applications of derivative: Surface and volume

A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 66 ft^3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.