Calculus Answers

A manufacturer wants to build a spring that takes a force 10 N (in negative direction) to compress it 0.2 m from the equilibrium position. The spring should be able to stretch 0.5 m from the equilibrium position.

As a mechanical engineer, you were asked to present how much work should be done to stretch the spring.The presentation shall include important components of the problem, complete and correct computations and a logical and organized explanation.

Hints:

First, create a force function F(x) by finding the spring constant k in F(x) = kx where F(x) is the force and x is the position from the equilibrium.

Then, calculate work using the concept of the definite integral.

If a variable force F(x) moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is W = ∫ ^b a F(x) dx.

A manufacturer wants to build a spring that takes a force 10 N (in negative direction) to compress it 0.2 m from the equilibrium position. The spring should be able to stretch 0.5 m from the equilibrium position.

As a mechanical engineer, you were asked to present how much work should be done to stretch the spring.The presentation shall include important components of the problem, complete and correct computations and a logical and organized explanation.

Hints:

First, create a force function F(x) by finding the spring constant k in F(x) = kx where F(x) is the force and x is the position from the equilibrium.

Then, calculate work ustng the concept of the definite integral.

If a variable force F(x) moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is W = ∫ ^b a F(x) dx.

Obtain a four-term Taylor polynomial approximation valid near x=,0 for each (1+x)^-1/3. Estimate the range of x over which three term polynomials will give two decimal accuracy

Prove that 𝑓(𝑥) = 𝑥^2 + 2𝑥 is not injective.

Find the directional derivative of f(x,y,z)=(x^2+y^2+z^2)^-1/2 at (3,1,2) on the direction of (yzi^ +xzj^ +xyk^)

A manufacturer wants to build a spring that takes a force 10 N (in negative direction) to compress it 0.2 m from the equilibrium position. The spring should be able to stretch 0.5 m from the equilibrium position.

As a mechanical engineer, you were asked to present how much work should be done to stretch the spring.The presentation shall include important components of the problem, complete and correct computations and a logical and organized explanation.

Hints:

First, create a force function F(x) by finding the spring constant k in F(x) = kx where F(x) is the force and x is the position from the equilibrium.

Then, calculate work ustng the concept of the definite integral.

If a variable force F(x) moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is W = ∫ ^b a F(x) dx

Let 𝑓(𝑥) = 1 3 𝑥 3 + 𝑥 2 − 15𝑥 − 9 . Use detailed sign tables in answering the following questions. (a) Find the intervals in which 𝑓 is increasing or decreasing. (b) Find the intervals in which the graph of 𝑦 = 𝑓(𝑥) is concave upwards or downwards. 

Air is escaping from a spherical balloon at the rate of 2𝑐𝑚3 per minute. How fast is the radius shrinking when the volume is 36𝜋 𝑐𝑚3 ?

Find the rate of change of the area 𝐴, of the circle with respect to its circumference C, 𝑖. 𝑒 𝑑𝐴 𝑑�

Consider the relation 𝑃 = {(1,7), (−1,7), (3,9), (1,3)}. Is 𝑃 a function? Justify your answer.

I. Investigate whether the following functions are even or odd:

(a) f(x) = x³

(b) f(x) = cos x

II. State the mean value theorem

III. (a) Find the derivative of the function y = 2x² + 12/x² when x = 2

(b) f(x) = -3/x-7. Find the inverse of the function.

Iv. Consider the function f(x) = e^rx Determine the values of r so that f satisfies the equation f"(x) + f'(x) - 6f(x) = 0.