Calculus Answers

Find dp/dq If P= q²+3/(q-1)³+(q+1)³

A spring is such that a 16 lb weight stretches it by 1.5 in. The weight is pulled down to a point

4 in below the equilibrium point and given an initial downward velocity of 4 ft/sec. An impressed

force of F(t) = 2 cos 74t is acting on the spring. Describe the motion.

Change the following point from polar to rectangular coordinate.

(3/2, n/12)

II. Evaluate f²(3ײ- 6x ‐2)dx

A small factory producing a single product has weekly fixed costs of production of $2,112 and weekly variable costs of $52x + 3/4 x2, where x is the quantity produced. the capacity of the factory is about 600 units.

Past experience suggests that the product’s price and quantity are linked by the following demand equation: p = 200 - 1/4 x (p, x > 0) where p = $ price/unit and x = quantity sold. You are required to:

(a) Find the level of production at which revenue is maximized

(b) Find any break-even points

Apply separation of variable to solve

x²u_xy+9y²u=0

A builder has 2400 feed of barrier and wants to barricade off a rectangular ground that borderlands a straight water flow. No barrier is required along the entire length of the water flow.

Find the dimension of the ground that has the largest area

The velocity of a moving object is given by the function

𝒅𝑽=𝟓𝒕^2+2t-4/t^2

a) Integrate the function and find the distance travelled in metres between t=2 and t=5

b) Calculate the distance travelled using a numerical method and compare your answer with part A.

01) Identify the X intercepts of this quadratic function y = 2x- x²

(02) Find the coordinates of maximum point of y = -4x + 8x – 2 by using equation y by using equetion method.

(03) Graph the quadratic function y = -3x +x+1 and you are required to answer the followings by using the graph:

(i) Coordinates of maximum point

(ii) Axis of symmetry

1) The extension, 𝑦, of a material with an applied force, 𝐹, is given by

𝒚 = 𝒆 𝑭×(𝟏×𝟏𝟎−𝟑) .

a) Calculate the work done if the force increases from 100N to 500N using:

i) An analytical integration technique

ii) Simpsons Rule

[Note: the work done is given by the area under the curve]

b) Compare the two answers

c) Increase the number of values used for your numerical method and analyse any affect the size of numerical step has on the result

2) Use numerical integration and integral calculus to analyse the results of a complex engineering problem.

The work done by a mechanism is given by:

𝒚 = 𝒙 𝐜𝐨𝐬 𝒙

a) Use integration by parts to determine the area under the curve between the limits of x = 7 and x =5 and hence the work done.

b) Choose a suitable strip width and use Simpsons rule to determine the area under the curve and hence the work done.

c) Evaluate the answers for a and b. Does one method verify the results of the other?

Find the local and absolute extreme values of the function on the given interval. Also

specify the intervals where function is increasing or decreasing

𝑓(𝑥) = 𝑥 + 2𝑐𝑜𝑠𝑥