Explore BrainMass


Derivatives and Limits

Find the derivative by the limit process. See attached file for full problem description. keywords: definition of the derivative, difference quotients

Derivatives and Rate of Change

(1.) A particle moves along the x-axis so that at any time that t is greater than or equal to zero, its position is given by x(t)= t^3-12t+5. a.) Find the velocity of the particle at any time t. b.) Find the acceleration of the particle at any time t. c.) Find all values of t for which the particle is at rest. d.) Find the s

Applying Derivatives

1.) Mean Value Theorem: Let f(x)= x ln x a.) Write an equation for the secant line AB where A= (a,f(a)) and B= (b,f(b)) b.) Write an equation for the tangent line that is parallel to the secant line AB. 2.) Approximating functions: Let f be a function with f'(x)= sinx^2 and f(0)= -1 a.) Find the linearization of f at x=0

Use the chain rule

Use the chain rule to find dx/dy for the next 2 problems 1. y=5u^2+u-1; u = 3x+1 2. y=square root of (u); u = x^2+2x-4

Derivatives, Difference Quotient and Equation of a Tangent Line

Compute the derivative of the function using the difference quotient and find the equation of the line that is tangent to its curve for x=c 1. f(x) = x^2-3x+2; c=1 2. y = (x+7)/(5-2x); c=1 Find the first and second derivative for for the next 3 problems 1. y=6x^5-4x^3-5x^2-1

Applications of Derivatives Word Problems : Maximizing Area and Revenue Functions

4. A Norman window consists of a rectangle with a semi-circle mounted on top (see the figure). What are the dimensions of the Norman window with the largest area and a fixed perimeter of P meters? 5. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each pers

Applications of Derivatives : Maximum Volume and Tangents

1 A box with its base in the xy-plane has its four upper vertices on the surface with equation z=48-3x^2-4y^2. What is the maximum possible volume. 2 Find the differential dw for w =ysin(x+z) 3 Find the equation of the plane tangent to z=-sin((pi)yx^2) at the point P =(1,1,0)

Estimates using Functions and Derivatives

5. The number of new customers for an internet business, y, in a month is a function of the number of advertising email announcements, x, that are sent out that month. So y=f(x). a) What is the meaning of f(1250)= 22 and f'(1250)= 0.06? b) Use the information in part (a) to estimate f(1300) and f(1050). Of the two estimates, w

Real Analysis: Derivatives and Sequences

Suppose that f: [a,b] &#61664; R is differentiable, that 0 < m f '(x) M for x &#1108; [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 &#1108; [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn &#1108; [a

Force as the Gradient of Potential Energy

See attached file for full problem description. 1) Find the partial derivatives with respect to x, y, and z of the following functions: (a) f(x, y, z) = ax2 + bxy + cy2, (b) g(x, y, z) = sin(axyz2), (c) h(x, y, z) = aexy/z^2, where a, b, and c are constants. 2) Find the partial derivatives with respect to x, y, and z of t

Derivatives, Differentiable Functions and Rate of Change

1. Functions f, g, and h are continuous and differentiable for all real numbers, and some of their values and values of their derivatives are given by the below table. x f (x) g(x) h(x) f'(x) g'(x) h'(x) 0 1 -1 -1 4 1 -3 1 0 3 0 2 3 6 2 3 2

Finding Derivatives and Logarithmic Differentiation

2. Find d y / d x : a) x^2 + xy &#8722; y^3 = x y ^2 b) sin ^ 2 y = y + 2 c) y = sqrt((x^2+1)/(x^2 - 5)) d) y = x^(ln sqrt(x)) 3. If x ^y = y ^x , use logarithmic differentiation to compute dy / dx at the point (3, 3).

Sunrise Baking Company markets dough nuts through a chain of food stores.

Sunrise Baking Company markets dough nuts through a chain of food stores. It has been experiencing overproduction and underproduction because of forecasting errors. The following data are its production in dozens of doughnuts for the past four weeks. Doughnuts are made for the following day; for example, Sunday's doughnut pro