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# Calculus and Analysis

### Dynamic Programming

For a computer to work properly, three subsystems of the computer must all function properly. To increase the reliability of the computer, spare units may be added to each system. It costs \$100 to add a spare unit to system 1, \$300 to system 2, and \$200 to system 3. As a function of the number of added spares (a maximum of two s

(a) Find the point of intersection of the line containing the points (3; 2; 1) and (4; 3; 3), and the plane containing the points (5; 4; 2), (3; 1; 6) and (6; 5; 3). (b) Find the radius of curvature at (1; 0; 1) on the three dimensional curve given by r(t) = cos ti + sin tj + e^tk. Sketch this curve, and briefly describe its

### The Centroid

Please help me with the following calculus problems: 1) Set up (do not integrate) an integral for the length of the curve y=tan-1x for x E [0,π). 2) Find the surface area obtained by rotating the curve x=2-y2 around the y axis. 3) Find the centroid of the region bounded by the curve x=2-y2 and the y axis.

### Effect of change in interest rate on price of a bond

Both Bond Sam and Bond Dave have 9 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 4 years to maturity, whereas Bond Dave has 15 years to maturity. If interest rates suddenly rise by 3 percent, the percentage change in the price of Bonds Sam and Dave is ??? percent and ??? percent, respecti

### Contrasting Ordinary and Partial Derivatives

Please also explain when we should use ordinary derivatives and when we should use partial derivatives.

### Exercises on Functions: Evaluating Linear Functions

1. Evaluate the function f(x) = 4x + 6 for x=4. 2. Evaluate the function f(x) = 9x - 6 for x=0. 3. Take a look at the following table: x -2 -1 0 1 2 f(x) -5 -2 1 4 7 a. Write out an equation for f(x). Assume the function is linear b. What is the slope? Is it negative or positive?

### Functions and Models

Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c. a. f(x) = x + 3 b. f(x) = 73 if x>2 otherwise f(x) = -1 c. f(x) = 79if x>0 or f(x) = -9 if x<0

### Analysis: Riemann Sums

(a) Given a bounded function fâ?¶[a,b]â?'R, and a partition Pâ?¶a=?_0<?_1<?_2<â?¯<?_n=b, Define the lower Riemann sum L(f,P) and the upper Riemann sum U(f,P) of f with respect to P, as well as the lower and upper integral of f. When is f called Riemann integrable and how is â?«^b_a f(x)dx defined in this case? (b

### Differential equation: Rate of change of quantity

A drug that has an half life of 20 hours inside the human body and is administered intravenously (I.V.) into a patient. The rate of change of the drug in the body is proportional to the amount present. Write a differential equation explaining the rate of change of the quantity of the drug and determine the constant proportion

### plane passing to the cuts

Find an equation of the plane that passes through the point (1,2,3) and cuts off the smallest volume in the first octant.

### Limits of square roots

Could you please show me how to calculate the examples attached? Thank you. 5. lim (x+1)/SQRT (10+x) - 3 x->-1 6. lim (x-1)/SQRT (x+3) - 2 x->1 7. lim (2x+7)(5x - 1)(12x + 2)/(2x - 7)^3 x->&#8734; 8. lim (3x+7)(4x - 1)(9x + 1)/(3x - 5)^3 x->&#8734;

### Maximizing an inscribed cylinder

1. A cylinder is inscribed in a right circular cone of height 5 and radius (at the base) equal to 8. What are the dimensions of such a cylinder which has maximum volume? Radius=________ Height=________

### Related Rate Ladder Problem is assessed.

A 16-ft ladder leans against a wall. The bottom of the ladder is 5 ft from the wall at time t = 0 and slides away from the wall at a rate of 3 ft/s. Find the velocity of the top of the ladder at time t = 1. a) The textbook response to this question is - sqrt (3) is approximately equal too -1.732 ft/s. The minus sign means

### all relative maximum locations

F(x)=-0.2x^4 + 3.7x^3 -7.4x^2 - 49.8x +68 1. find all x intercepts 2. all relative minimum location (order pairs) 3. all relative maximum locations (order pairs) 4. increasing 5. decreasing 6 range (round to the nearest hundredth when necessary)

### Oblique or Slant Asymptote

Find the oblique (or slant) asymptote of y= (3x^(3) + 6x^(2))/3x^(2) -3x + 4 and express your answer in the form y=mx+b where m and b are constants. Answers: The oblique asymptote is y=______x +_______

### critical point(s) and its absolute extrema

For the following function, find all its critical point(s) and its absolute extrema. [ Note: If there is more than one answer enter separated by commas.] f(x) = 21 sqrt(x^(2)+38) - 12x, 0 < [or equal to] x < [or equal to] 14. Answers: The critical point(s) of f(x) is/are x=_____ f attained its absolute maximum

### Finding the Critical Point and Relative Minimum and Maximum

Consider the function f(x) = 4kx^(3) - (k^2 - 13)x + 9k, where k is a constant. (a) Suppose that f has a critical point at x = 1. Find all possible value(s) of k. If there is more than one answer, enter them separated by commas. Answer: f has a critical point at x = 1 exactly when k = ______ (b) If f has a relative max

### Calculus problem

Let f(x)=x^(4)â?'6x^(3)+12x^(2). Find (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points. (a) f is increasing on the interval(s) = (b)

### Revenue Maximization

Demand for a certain product is represented by the following equation: p=(80-q)/4; 0<=q<=80 Where q is the # of units and p is the price per unit. At what quantity sold will the revenue be maximized?

### Calculating Distance: Ship Example

1. (1 pt) At noon, ship A is 180 km west of ship B. Ship A is sailing south at 20 km/h and ship B is sailing north at 40 km/h. How fast is the distance between the ships changing at 4:00 PM? _________km/h

### method of logarithmic differentiation

Let f(x) = (x^(3x) (4x-1)^8)/sqrt(8x^(2)+10) If we use the method of logarithmic differentiation to find f '(x), we obtain f '(x) in the form f ' (x)=[(x^(3x) (4x-1)^8)/sqrt (8x^(2)+10)] [(A(x) + (B(x)/4x-1) + (C(x)/8x^(2)+10)] where A(x), B(x) and C(x) are functions of x. Find them. A(x)=______ B(x)=______ C(x)=_

### Calculus: Finding the Derivative

In each of the following, differentiate the function and simplify your answer into the specified form. (a) g(x)=x(4sin3x+6cos3x). Answer: g'(x)=(Cx+D)sin3x+(Ex+F)cos3x with constants C=_______,D=_______,E=_______, and F=______ (b) f(x)=x^(7) (7x-4)^5 f ' (x)=_________, which can be simplified to f ' (x)= x^(6) (7

### Cauchy-Riemann equations

Complex Differentiation 1) Suppose that an analytic function f defined on the whole of C satisfies Re(f(z))=0 for all z in C. Show that f is constant. 2i) Verify that u=x2-y2-y is harmonic in the whole complex plane. 2ii) Suppose f(x,y)=u(x,y)+iv(x,y). The Cauchy-Riemann equation state that: ux=vy and uy=-vx. For u=x2-

### Rate of Change Example Problem

The average number of vehicles waiting in a line to enter a parking ramp can be modeled by the function f(x) = (x^2) / [2(1-x)] where x is a quantity between 0 and 1 known as the traffic intensity. Find the rate of change of the number of vehicles in line with respect to the traffic intensity for the following values of th

### Calculus question for differentiated given functions

Differentiate the given functions. For this question, you may use the following differentiation rules and formulas: If r is any rational number, then d/dx (x^r) = rx^(r-1) If f and g are differentiable functions of x, and if ? [alpha] and ? [beta] are constants, then d/dx ( ?f(x) + ?g(x)) = ? d/dx (f(x)) + ? d

### Kinematics: Determining Velocity

Bob the Iguana is waiting to use his Giant Slingshot to shoot the Evil Bart the Blackbird out of the sky as he flies overhead. Bob's Giant Slingshot launches a stone vertically so that the function h(t) = 300t — 16t^2 models the height h in feet of the stone t seconds after it leaves the slingshot. Answer each of the following

### Pre-calculus Word Problems

1) Jack and Jill ran a 100 meter race. Jill ran the race in 10 seconds and won by 5 meters; Jack had run only 95 meters when Jill crossed the finish line. They decide to race again, but this time Jill starts 5 meters behind the starting line. Assuming that both runners run at the same pace as before, who will will the second rac

### evaluate the hemoglobin function

Problem 10.6. Myoglobin and haemoglobin are oxygen carrying molecules in the human body. Hemoglobin is found inside red blood cells, which flow from the lungs to the muscles through the bloodstream. Myoglobin is found in muscle cells. The function calculates the fraction of myoglobin saturated with oxygen at a given pressu

### Growth and decay models

When a vertical beam of light passes through a transparent medium, that rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear sea water, the intensity 3 feet below the surface is 25% of the initial intensity I0 of the incident beam. What is the inten

### Steps to solve a limit question are noted.

1. lim (12x^(3) + 12)/5x^(4) + 5x^(3) + 7 x-> infinity 2. lim (12x^(3) + 12)/5x^(4) + 5x^(3) + 7 x-> -infinity 3. lim (3x^(3) - 9x^(1) + 7)/3x^(3) + 7x^(2) - 7 x->infinity 4. lim (3x^(3) - 9x^(1) + 7)/3x^(3) + 7x^(2) - 7 x-> - infinity 5. lim