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# Calculus and Analysis

### 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

### 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)

### 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

### 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

### Identify the type of sampling

1: Given the situation described, identify the type of sampling involved. A student doing a report on fashion trends asks all her best friends their opinion on plaid clothing. - Random sampling - Systematic sampling - Convenience sampling - Cluster sampling - Stratified sampling 2: Scripps Hospital surveyed 10 patients

### Tangents and Normals

Find the tangents and the normals at any point of the following curves: 1) y=1/x^2+2 (1,1/2) 2) y=6/(x^2+1)^2 (1,3/2)

### Calculus and the Connecticut River

Needed info: Whenever the Connecticut River reaches a level of 105 feet above sea level, two Northampton, Massachusetts flood control station operators begin a round the clock river watch. Every two hours they check the height of the river using a scale marked off in tenths of a foot, and record the data in a log book. In

### Solving Differential Equations with given initial conditions

Please see attached Please show all work with explanations. Differential Equations 5. Solve the differential equation with the given initial conditions: 6. Solve the differential equation with the given initial conditions:

### First-Order Differential Equations

Two Snowplows - Differential Equations (First-Order Differential Equations) One day it began to snow exactly at noon at a heavy and steady rate. A snowplow left its garage at 1:00pm, and another one followed in its tracks at 2:00pm. a)At what time did the second snowplow crash into the first? To answer this question, assum

### Calculas

Please show all woCarlos is blowing air into a soap bubble at the rate of 7 cm3/sec. Assuming that the bubble is spherical, how fast is its radius changing at the instant of time when the radius is 11 cm? cm/sec. How fast is the surface area of the bubble changing at that instant of time? cm2/sec. The Millers are plann

### Differential Equations Function for Region Bounded

1. Solve the following differential equation: -2yy' +3x² √(4-y²) =5x² √(4-y²) , -2< y<+2 2. A calculus instructor has determined that the arc of an individual diving into a swimming pool is defined by the function, f(x) = sin (.4x). Determine how far the diver has traversed in his dive as he passes thr

### Tax Formula: How much in taxes is paid by individuals with incomes of \$10,000, \$30,000, and \$50,000? What are the average tax rates for these income levels? At what income level does tax liability equal total income?

Here's the problem I'm having trouble with: Taxes in Oz are calculated according to the formula T=.01I^2 where T=thousand of dollars of tax liability and I=income measured in thousands of dollars. How much in taxes is paid by individuals with incomes of \$10,000, \$30,000, and \$50,000? What are the average tax rates for thes

### Calculating Income Based on Salary and Commission

Please help with the given problems including all the steps used to achieve the solution. Isaac earns a base salary of \$1250 per month and a graduated commission of 0.4% on the first \$100,000 of sales, and 0.5% on sales over \$100,000. Last month, Isaac's gross salary was \$2025. What were his sales for the month?

### Conducting Various Probability Calculations

1. How many 8 letter arrangements can be made from the word RECYCLE? 2. In an experiment, a student is randomly selecting a letter from one of 4 boxes after selecting the first box at random. What is the probability the student will select the letter A? Box 1 has letters S,O,H Box 2 has letters S,O,H Box 3 has letters C,A,H

### Differentiate Introductory Calculus

1. Consider the function f(x)=(&#8722;5x^(2))+7x+15. a) For the given function f, the expression f(x+h)-f(x)/h can be simplified to the form Ax+Bh+C, where A, B and C are constants. Do it. Answer: f(x)=(&#8722;5x^2)+7x+15=____x+____h+____ b) Using the result of (a), find the derivative of f at x . Answer: f'(x)=______.

### Simple quick intro calculus questions

1) Let f(x)= {-x+b, if x<-1 {5, if x=-1 {(-5/(x-b))+4, if x>-1 (and x=b) a) For what value(s) of b in f continuous at -1? b=________ b) For what value(s) of b does f have a removable discontinuity at -1? b=________ c) For what value(s) of b does f have an infinity discontinu

### Interval Notation

1) Solve the following inequality. Write the answer in interval notation. X/X+3 ≤ -2/5(X-3) 2) Solve the inequality. 3 < |5X+3| < 5 3) a)Find an equation of the line, say y=mx+b, which passes through the point (-3,8) and is perpendicular to the line 3x+3y=33. y=? b)What is the shortest distance from the point (

### Fundamental Theorem of Calculus and Uniform Convergence

Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded) Let F(x) = {█(x^2 sin⁡(1/x) if 0<|x|≤1,@0 if x=0)┤ and let f(x) = F'(x) Prove that F'(x) exists for all x ∈[-1,1] Find f(x) for all x ∈[-1,1], and prove t

### Autonomous differential Equation

The rate at which a bacteria population multiplies is proportional to the instantaneous amount of bacteria present at any time. The mathematical model for this dynamics can be formulated as follows: db/dt = kb where b is a function in terms of the time t, b(t) is the number of bacteria at the time t, and k is a constant.

### Differential Equation Word Problem: A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and ...

A body falling in a relatively dense fluid, oil for example, is acted on by three forces: the weight W due to gravity (acting downwards), a resistence force R and a bouyant force B (both actin upwards). The wieght W of the object of mass m is mg. The bouyant force B is equal to the weight of the fluid displaced by the object.

### Piecewise Function Straight Lines and Parabola

4. Solve: a Solve for x if 2logx = 2 + log25 b Simplify log3 81 - log3 27 + 4log3 v3 c Find the inverse of the function f(x) = e^x + 1 5. This is completing a piecewise function which consists of 2 straight lines and a parabola. Coordinates given are (-6,0) (-3,6)-straight line, (-3,6) (.5, -6) (2,-4) -parabola, and last

### Mathematics

1.Find the values of a and b for the polynomial f(x) = 2x^3 + ax^2 - 4x + b, given that f(x) is divisible by x+1 and x-3. Write f(x) in a factorised form. Hint: consider f(-1) and f(3) 2. The population of a country is found to be growing continuously at an annual rate of 1.98% t years after 1 January 1960. The total populat

### Find the slope of the function's graph at the given point

1. Find an equation for the line tangent to y= -5-5x^2 at (5,-130) 2. Find the slope of the function's graph at the given point. Then find an equation for the line tangent to graph: f(x) = x^2 + 2, (-2,6) What is the slope of the function's graph at the given point? 3. Using the defenition, calculate the derivative

### Calculus and Inequalities in Word Problems

Calculus applications Exercise 6.1 23. Manufacturing A company manufacturing two types of leaf blowers, an electric Turbo model and gas-powered Tornado model. The company's production plan calls for the production of at least 780 blowers per month. a. Write the inequality that describes the production plan, if x represents

### Calculus questions

1. What is missing in this surface of revolution as a function of x revolved around the y-axis: ? A) Nothing, the formula is fine as is. B) Nothing, but the limits of integration should be on the interval [c, d]. C) The variable x should be x2. D) The constant 2&#960; is missing. 2. Find the solid of revolu

### Supply and Demand Functions in Equilibrium

Consider a market with the following supply and demand functions: QD = a0 - a1PD a0, a1 > 0 QS = b0 + b1PS b0, b1 > 0 (a) Find the equilibrium quantity and price as a function of the parameters (use any method you like). Are there any additional restrictions that you must impose on the parameters for thi

### Determining the Maximum Separation

The positions of two particles on the s-axis are s1=sin t and s2=sin(t+ π /3), with s1 and s2 in meters and t in seconds. a) At what time(s) in the interval 0 ≤t ≤ 2π do the particles meet? b) What is the farthest apart that the particles ever get? c) When in the interval 0 ≤t ≤ 2π is the distance between the parti

### derivative of the function ..

1. Compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value of the independent variable. (Compute the derivative of the function from the definition only, using limits. More advanced methods are not allowed here. Show your work.) (i) f(x) = x^2 - 1 x = -

### The Manager of a company that produces graphing calculators determines that when x thousand calculators are produced, they will all be sold when the price is p(x)=1,000/0.3x^2+8 dollars per calculator. A. At what rate is demand p(x) changing with respect to the level of production x when 3,000(x=3) calculators are produced? B. The revenue derived from the sale of x thousand calculators is R(x)=xp(x) thousand dollars. At what rate is revenue changing when 3,000 calculators are produced? Is revenue increasing or decreasing at this level of production?

The Manager of a company that produces graphing calculators determines that when x thousand calculators are produced, they will all be sold when the price is p(x)=1,000/0.3x^2+8 dollars per calculator. A. At what rate is demand p(x) changing with respect to the level of production x when 3,000(x=3) calculators are produced?