Share
Explore BrainMass

# Derivatives

### Speed of Plane/Boat

See attached file for full problem description. 30. A boat is pulled into a dock... (a) Determine the speed of the boat when there is 13 feet of rope. What happens to the speed of the boat as it gets closer to the dock? (b) Determine the speed of the rope when there is 13 feet of rope. What happens to the speed of the ro

### Derivatives and Limits

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

### Applications of Derivatives and Rate of Change

1.) (d/dx)(xe^(lnx^2))=? 2.) If x=e^(2t) and y=sin(2t), then (dy/dx)=? 3.) If y=xy+x^2+1, then when x=-1, (dy/dx) is ? 4.) A particle moves along the x-axis so that its acceleration at any time is a(t)=2t-7. If the initial velocity of the particle is 6, at what time t during the interval 0&#8804;t&#8804;4 is the particl

### 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

### Logarithmic Differentiation

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

### 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)

### 1. Find the point on the graph of y= e^x at which the curvature is the greatest 2. equation of the surface by revolving the curve 3. cylindrical coordinate equation of the surface

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.

### 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

### Differentiation for a Tangent Line

Differentiation 1. Show that if the tangent to y=ekx at (a, eka) passes through the origin then a=1/k. 2. Find the value of a and b so that the line 2x +3y = a is tangent to the graph of f(x)=bx2 at the point where x = 3. See attached file for full problem description.

### 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

### 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

Y= x/(e^x +1)

### Chain rule?

Please explain chain rule i'm a visual person please show several examples

### Quotient and Composite Rules

1. a. use quotient rule to find derivative of this function. f(x) = (20+16x-x^2)/(4+x^2). b. Find any stationary points of the function from 1a. An use the first derivative test to see whether they are local maximum or local minimum of f(x). c. what are the maximum and minimum values of the function f(x) at interval [-6,2]

### Find derivative

Two problems are included in attachment.

### How fast is the brick falling after 2 seconds have passed?

A brick comes loose from near the top of a building and falls such that its distance s (in feet) from the street (after t seconds) is given by the equation s(t) = 200 - 16t^2 (see equation in attached file) How fast is the brick falling after 2 seconds have passed?

### Finding Derivatives (12 Problems)

Answers and working to the questions: 1. Obtain dy for the following expressions. dx (a) y = (5x + 4)3 (b) y = (3 - 2x)5 (c) y = square root (5 - 0.6x) (d) y = (2 + 3x)-0.6 2. Differentiate the following with respect to o. (a) f(o) = sin(5o - 2) (b) f(o) = cos(4 - 3o) (c)

### Derivatives and Rate of Change of Curves

See the attached file. 1: Both forms of the definitions of the derivative of a function f at number a. 2: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2ft/sec how fast will the foot of the ladder be moving away from the wall when the top is 5ft above the ground? 3: y':

### Find the first three derivatives of the function f(x) = 2cos x sin 2x.

Find the first three derivatives of the function f(x) = 2cos x sin 2x. (See attached file for full problem description)

### Finding derivative of a rational function using Chain rule or determining du/dx of a rational function

Calculus - Chain Rule. Finding Derivatives using Chain Rule. See attached file for full problem description.

### Derivatives

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

### Finance : Derivatives, Tangents, Interpreting the Slope of a Regression Line and Revenue Functions

1. The table below presents the net sales (Revenue), R(t) in billions of dollars for Wal-Mart for the period 1994 to 2004 (Wal-Mart's website). Let t = 0 represent 1990. t 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 P(t) 63 78 89 100 112 131 156 181 204 230 256 a. Use your graphing utility to find the reg

### Applications of Derivatives : Minimize Cost by Minimizing Volume of a Cylindrical Can

We've got a cylindrical can with height =h and radius =r. It will hold 4L (4,000 cm cubed) of some liquid. The material for the top and bottom costs 2 cents per square cm and the material for the side costs 1 cent per square cm. Find h and r to minimize the cost. keywords: derivative, differentiation, differentiate, mini

### 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,