### Derivatives and Limits

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

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

(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

Find the derivative: f (x)= ln[x(x^2+5)/sqrt(x^3-5)]

Use logarithmic differentiation to find dy/dx: y=(x+2)*sqrt[1-x^2]/(4*x^3)

Using the chain rule, show that if z = y + f(x^2 - y2), where f is differentiable, then partial derivatives y(dz/dx) + x(dz/dy) = x

Prove the following: If f(x) = 1 if x is greater than or equal to 0 and f(x) = 0 if x < 0, then there is no function F such that F'(x) = f(x) for every x in R. See #9 in the attached file.

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 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

Find dy/dx by implicit differentiation 1. (3x^2+1)^4 for this problems find d^2y/dx^2 1. y/x - x/y = 5

Find the first and second derivative for for the next 2. z=2/(1+x^2) 3. f(x) = (x-1)/(x+1^2) find dy/dx by implicit differentiation 1. square root of (2x) + y^2 = 4

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

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

7. b) (fg)' = f'g + fg' c) (f/g)' = (f'g - fg') / g^2

Q: Find the derivative of the following function h(x) = x^5/(x^2-7). Please view Word Doc for a clearer version.

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)

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

1 Find the point on the graph of y= e^x at which the curvature is the greatest. 2 Write the equation for the surface generated by revolving the curve x^2 - 2y^2 = 1 about the y-axis. Describe the surface 3 The parabola z = y^2, x =0 is rotated around the z-axis. Write a cylindrical-coordinate equation for the surface.

Investigating conic sections - you will be required to draw graphs from given information, describe features of graphs and find equations to fit given graphs.

Suppose that f: [a,b]  R is differentiable, that 0 < m f '(x) M for x є [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 є [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn є [a

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

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

2. Find d y / d x : a) x^2 + xy − 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).

See attached file for full problem description.

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

See attached file for full problem description.

"Define f(x) to be the distance from x to the nearest integer. What are the critical points of f."

Find any relative extrema of the fx f(x) = arcsin x - 2x

Find derivative h(x) = x^2 arctan x

Find the derivative of the f(x) f(t) = arcsin t^2

G(x)= ln(cosh x)