Calculus Answers

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

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

specify the intervals where function is increasing or decreasing

(i) 𝑓(𝑥) = (𝑥²+ 𝑥 + 1)²

Q: 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) = 𝑥²e^-x

The equation for a displacement 𝑠(𝑚), at a time 𝑡(𝑠) by an object starting at a displacement of 𝑠0 (𝑚), with an initial velocity 𝑢(𝑚𝑠 −1 ) and uniform acceleration 𝑎(𝑚𝑠 −2 ) is: 𝑠 = 𝑠0 + 𝑢𝑡 + 1 2 𝑎𝑡 2 A projectile is launched from a cliff with 𝑠0 = 30 𝑚, 𝑢 = 55 𝑚𝑠 −1 and 𝑎 = −10 𝑚𝑠 −2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑡) for the first 10s of motion. b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2 ) d) Use your results from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).

A delivery company accepts only rectangular boxes the sum of whose length and the perimeter of a cross-section does not exceed 108 inches. Find the dimensions of an acceptable box of largest volume. 

The edge of a cube was found to be 30 cm with a possible error in measurement of .1 cm. Use differentials to estimate the percentage error (to the nearest hundredth) in computing (a) the volume of the cube and (b) the surface area of the cube.

 An open box is to be made out of a 8-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

A rectangular billboard 5 feet in height stands in a field so that its bottom is 6 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x

x

 feet from the billboard. Express θ

θ

, the vertical angle subtended by the billboard at her eye, in terms of x

x

. Then find the distance x

x

 the cow must stand from the billboard to maximize θ

θ

.