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# Derivatives

### Directional Derivative

Find the directional derivative of f(x,y) = 2x^3-y^2+xy at the point (1,2) in the direction of the vector (1,3). Be careful: That direction vector isn't a unit vector!

### This problem demonstrates the technique of implicit differentiation.

Differentiate the following.... x^2-4xy+3ysinx=17.

### Multivariable Calculus : Triple Integral - Cylidrical Coordinates

Solve by triple integration in cylindrical coordinates. Assume that each solid has unit density unless another density function is specified: Find the volume of the region bounded above by the spherical surface x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 + y^2

### Using Dynamic Programming to Solve Problems

Please see the attached file for the fully formatted problem. Use Dynamic Programming to solve: 1. Min f(x-bar) = 3x21 + x22 + 2x23 s.t. Sx1 + 2x2 +x3 >= 18 DP Formulation:.... Min s.t. Stage 1: Stage 2: Stage 3:

### Working with orthogonal trajectories

(a)Find the orthogonal trajectories of the family of curves defined by 2cy + x2 = c2, c>0 State the differential equation of the orthogonal family, and show your steps in obtaining a solution. (b) On the same set of "square" axes, plot at least five members of each of the given family and your family of orthogonal soluti

### Derivative of a function problem

Let f be the function whose graph goes through point (3,6) and whose derivative is given by f'(x) = (1+e^(x))/(x^2) a) write the equation of the line tangent to the graph of f at x=3 and use it to approximate f(3.1) b) Use Euler's method, starting at x=3 with a step size of .05 to approximate f(3.1). Use f'' to explain wh

### Partial derivative question

I am taking a course by distance, and my professor provided an example of how to create a Hessian matrix using partial derivatives. He gave another example that just had the solution for us to try on our own. I think that I am somehow not taking the second order partial derivative right. The attached file has the professor

### Derivatives: Minimizing Time and Distance

A lighthouse is 30 miles off a straight coast and a town is located 25 miles down the sea coast. Supplies are to be moved from the town to the lighthouse on a regular basis and at a minimum time. If the supplies can be moved at a rate of 4 miles per hour on water and 40 miles per hour on land, how far from the town should the do

### Partial derivatives

The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation: &#8706;u/&#8706;t = (c^2)&#8706;^2u/&#8706;x^2 where t is the time, x distance from the beginning of the rod and c is the material constant. Function u(t,x) represents the temperature at the given time t and p

### Differentiation

Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial

### Maclaurin Series Hyperbolic Tangent

Given The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.

### Calculus : Partitions and Riemann Integral

Let f be the function: F(x) 1/4 x=o, x 0<x<1 3/4 x=1 Using standard partition Pn (0,1) where n greater or equal to 4 L(f, Pn) = 2n(squared) -3n+4 all divided by 4n(squared) U(f,Pn) = 2n(squared) +3n+4 all divided by 4n(squared) and deduce that f is intergrable on (0,1) and evaluate (intergral sign with 1 at t

### Calculate the derivatives of wave equations

The equation for a wave moving along a straight wire is: (1) y= 0.5 sin (6 x - 4t) To look at the motion of the crest, let y = ym= 0.5 m, thus obtaining an equation with only two variables, namely x and t. a. For y= 0.5, solve for x to get (2) x(t) then take a (partial) derivative of x(t) to get the rate of change of

### Implicit differentiation : the chain rule and product rule.

Use implicit differentiation to find dy/dx if y^2 + 3xy + x^2 + 10 = 0 (1) where y is a function of the independent variable x.

### Equation of a straight line which is tangent to a curve

For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1. See attachment for diagram.

### Using Cramer's Rule with a 3x3 System

Using Cramer's Rule with a 3x3 system 3x+4y+z=17 2x+3y+2z=15 x+y =4

### Definition of derivative

Let f(x) be a continuous function of one variable. a) Give the definition of the derivative. b) Use this definition to find the derivative of f(x)=x^2+2x-5 c) Evaluate f'(2)

### Dormitory Requirements for Students

Six students need to be placed in a dormitory. There are four double rooms, two single rooms, and two students cannot be placed together, how many ways are there to place the students?

### Proof Regarding a Twice Differentiable Function

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. Question: Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b]

### Maximizing the area of a rectangle.

Find the dimensions of the rectangle of the largest area that has its base on the x axis and its other two vertices above the x axis and lying on the parabola y=8-x^2.

### Absolute Maximum and Minimum of a Given Function

Find the absolute maximum and absolute minimum values of f on the given interval. F(x) = sqrt(9-x^2) [-1, 2] or in other words: F(x) equals the square root of (9 minus x squared). The problem is also attached in MS word.

### Using related rates to answer a pulley question.

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q? Express solution using related rate n

### Limits and derivatives

Find the derivative. a) f= 4-sqrt(x+3) b) f= (x+1)/(2-x) See attachment below for additional information.