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v(t) = 3 Cos(πt) − 2Sin(πt) (Eq. 1)
2- Using a mathematical model and calculus methods (e.g. numerical and
integration methods) to solve given engineering problem (Eq. 1).
Your tasks is
c) Find a mathematical model (e.g. equation) to correlate position and
time using an Excel sheet and trendline.
d) Using definite integration and driven equation (from c) to find the area
under curve over the time interval 0 ≤ t ≤ 3 seconds and C=12.
e) Using a mid-ordinate rule and driven equation (from c) to find the area
under curve over the time interval 0 ≤ t ≤ 3 seconds at h= 0.5.
f) Find an accurate mathematical model (e.g. equation) to correlate
position and time. To complete this task you should be able to sketch
the graph again, find the accurate equation using an excel sheet and
trendline.
.
The annual profit for a firm depends upon the number of units produced. the function which describes the relationship between profit P (stated the number of units produced x is
P-0.01x²+5,000x - 25,000
(a) Determine the number of units x which will result in maximum profit. (b) What is the expected maximum profit?
The demand function for a firm's product is
q-150,000-75p
where q equals the number of units demanded and p equals the price in dollars. (a) Determine the price which should be charged to maximize total revenue.
(b) What is the maximum value for total revenue? (e) How many units are expected to be demanded?
The curve y=2x^3-3x^2-36x+15 has a turning point at (3, -66) and (-2,59). Identify whether each of these points is a maximum or a minimum. (show all your work).
A curve has the equation y=3x^2-5x+4. Find the gradient of the curve at the point (2,6)
Q : 1 The distance x meters moved by a partide int seconds is given by x = t + 3t + 4 . Find the velocity and accelerating after 3 seconds .
Q : 2 The radius of a circle is increasing uniformly at the rate of 3 cm per second . Find the rate at which its area is increasing when radius is 10cm .
Q : 3 If displacement is s = sin2t , find , its acceleration .
(a) Let In = \int _0^1x^n\sqrt{1-x^2\:dx}, n ∈ N. Show that (n + 2)In = (n − 1)In−2, n ≥ 2.
(b) If g = sin(sin x), prove that \frac{d^2y}{dx^2}+tan\:x\frac{dy}{dx}+ycosx^2= 0.
1.By writing 3 cos x + 4 sin x = λ d dx (4 cos x + 5 sin x) + µ(4 cos x + 5 sin x), where λ and µ are constants
1. (a) Express 5 sinh x + cosh x in the form Aex + Be−x , where A and B are integers.
(b) Solve the equation 5 sinh x + cosh x + 5 = 0, giving your answer in the form ln a, where a ∈ R.
1. (a) Use the Mean Value Theorem to establish the following inequality
e a (x − a) < ex − e a < ex (x − a), if a < x.
