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Find the area bounded by the curve

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Reference no: EM132128222

ANALYTICAL METHODS FOR ENGINEERS ASSIGNMENT - CALCULUS

This assignment assesses Outcome - Analyse and model engineering situations and solve problems using calculus.

Questions -

Q1. Differentiate the following functions with respect to x.

(a) Use the quotient rule to differentiate

y = (2x4 - 3x)/(4x - 1)

(b) Use the Chain rule to differentiate

y = 6cos(x3 + 3)

(c) Select an appropriate rule to differentiate

y = (4x2 - e2x)sin3x

Q2. The angular displacement, θ radians, of the spoke of a wheel is given by the expression θ = 0.5t4 - t3, where t is time in seconds. Find:

(a) The angular velocity after 2 seconds.

(b) The angular acceleration after 3 seconds.

(c) The time when the angular acceleration is zero.

Q3. From a rectangular sheet of metal measuring 120 mm by 75mm, equal squares of side x are cut from each of the four corners. The remaining flaps are then folded upwards to form an open box.

(a) Draw a neat and simple diagram of the rectangular sheet of metal (not drawn to scale) and show the dimensions given including the squares of side x.

(b) Show that the volume of the box is given by: V = 9000x - 390x2 + 4x3

(c) Find the value of x such that the volume is a maximum.

Q4. Find the following integrals.

(a) ∫(5x2 + √x - 4/x2)dx

(b) ∫[cos(x/2) - sin(3x/2)]dx

(c) 1∫5 s/√(s2+4) ds

Q5. (a) Find the area bounded by the curve y = 1/x between x = 2 and x = 6. (Give your answer to 3 decimal places.)

(b) Sketch the curve for y = sin(x) between x = 0 and x = 2π, then find the total area enclosed by the curve y = sin x and the X axis between x = 0 and x = 1.7π.

Q6. If the instantaneous rate of change of a population is 50t2 - 100t3/2 (measured in individuals per year) and the initial population is 25000 then

(a) What is the population after t years?

(b) What is the population after 25 years?

Q7. Use integration by parts to solve the following integral

∫5xcos(4x)dx

Reference no: EM132128222

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Reviews

len2128222

10/1/2018 10:32:55 PM

Before you start please read the following instructions carefully. This assignment forms part of the formal assessment for this module. If you fail to reach the required standard for the assignment then you will be allowed to resubmit but a resubmission will only be eligible for a Pass grade, not a Merit or Distinction. You should therefore not submit the assignment until you are reasonably sure that you have completed it successfully. Seek your tutor's advice if unsure.

len2128222

10/1/2018 10:32:49 PM

Ensure that you indicate the number of the question you are answering. Make a copy of your answers before submitting the assignment. Complete all details on the front page of this TMA and return it with the completed assignment including supporting calculations where appropriate. Your tutor’s comments on the assignment will be posted on Blackboard.

len2128222

10/1/2018 10:32:44 PM

Select the appropriate rule to produce the correct answers. Produce a substantially correct solution by applying differentiation. Produce a correct solution using logical mathematical statements. Use an appropriate technique to produce correct solutions and in a neat and correct manner. Use the limits accurately to find the required area. Be able to apply calculus to produce a substantially correct solution. Solve the integral by applying the given rule and present the solution in a logical and coherent manner.

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