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Calculus and Analysis

Derivative Question - Algebraic Function

This is from a Trig/Calculus course...Explain FULLY: If F(x) = x^4 - 2x^3 + 4x^2 - 9 Note: ^ indicates exponant. Find F prime of x. It will be a derivative. I need every step explained clearly as I have a bet riding on this! I need to be able to show every step in order to win my bet.

Laplace Transform

If L[f(t)]=F(s) then L[t*f(t)]= -dF/ds use this result to compute L[t*e^kt].

Laplace Transform

Use the laplace transform to solve the ODE y"+3y = cos(2t), y(0)=0 , y'(0)=0 Show all details related to using the inverse transform.

Differential Equations

The birth rate in a state is 2% per year and the rate is 1.3% per year. The population of the state is now 8,000,000. a) At what rate are babies being born in the state now? with units b) At what rate are people dying in the state now? c) Write a differential equation that the population of the state satisfies. include

Advanced Calculus: The Mean Value Theorem and Directional Derivatives

Please see the attached file for the fully formatted problems. Let F: R^n --> R be continuously differentiable. Show that at each point x E R^n there is a direction hx so that the directional derivative is 0, i.e., df/dhx (x) = 0. Is hx unique? Give a method for determining hx.

Simulation : Skydiver in Free-fall

A skydiver, weighing 70kg, jumps from an aeroplane at an altitude of 700 metres and falls for (T) seconds before pulling the rip cord of his parachute. A landing is said to gentle if the velocity on impact is no more than the impact velocity of an object dropped from a height of 6 metres. The distance that the skydiver falls d

Lagrange Multipliers

Use Lagrange multipliers to determine the smallest value of the function f(x,y) = 2x^2-x+3y^2 for points (x,y) on the circle x^2 + y^2 = 1.

Setting up a Riemann Sum

I can't figure out exactly how to formulate a riemann sum. For example, when given y=x+2; [0,1], and told to "find the area of the region under the curve y=f(x) over the interval [a,b]. To do this, divide [a,b] into n equal subintervals, caluculate the area of the cooresponding circumscribed polygon, and then let n go to infin

Calculus : Reimann Sum and Limits and Continuity

34) The function f is continuous on the closed interval [1,5] and has values that are given in the table below. If 2 subintervals of equal length are used, what is the midpoint Reimann sum approximation of integral with 5 on top and 1 on bottom f(x)dx? Please given step by step explaination and answer is 32. x 1 2 3

Calculus : Antiderivative and Rate of Change

36)If the functions f and g are defines for all real numbers and f is an antiderivative of g, which statements are not true? I If g(x)>0 for all x, then f is increasing. II If g(z)=0 then f(x) has a horizontal tangent at x+a. III If f(x)=0 for all x, then g(x)=0 for all x. IV If g(x)=0 for all x, then f(x)=0 for all x.

Calculus : Differentiability and Maximizing Area

24) Let f be a differentibale function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that. I f'(c)=0 II f'(x)>0 when a<or equal to x<c, and III f'(x)<0 when c<b<or equal to b. Which is true? Then tell why others false. a. F'(c)=0 b. F"(c)=0 c. F(c) is an abs. ma

Calculus : Velocity, Continuity, Limits, Differentiability and Integrability

26) The verticle height in feet of a ball thrown upward from a cliff is given by s(t)=-16t^2+64t+200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero? 27)If the function f is continuous for all real numbers and the lim as h approaches 0 of f(a+h)-f(a)/h = 7 then which state

Critical Points and Maximum and Minimum Values

Please see the attached file for the fully formatted problem. Find the critical points and use your test of choice to give local maximum and minimum values. Give those values. f(x) = x^2 /sqrt(x^2 + 4)

Find dy

Please see the attached file.