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Problem of Optimisation: A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible.

First, draw a diagram showing the rectangle, the circle and the unknowns . Then find the equations representing this problem (equation of a circle of center 0, x 2 +y 2 =4 2 (1) and of the area of the rectan...
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How do you integrate ln(x) ?

-First write the integral with ln(x) and form a product of 1 next to it-Let u=ln(x) and differentiate this-Let dv/dx= 1 and integrate this-Substitute values into integration by parts formula-Simplify and add...
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Find the first derivative of f(x). f(x) = ln(3x^2+2x+1)

The differential of a natural log function gives a fraction where the numerator is the first derivative of what we are taking the natural log of and the denominator is what we are taking the natural log of. ...
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A curve C has equation y = x^2 − 2x − 24 x^(1/2), x > 0 (a) Find (i) dy/d x (ii) d^2y/dx^2 (b) Verify that C has a stationary point when x = 4 (c) Determine the nature of this stationary point, giving a reason for your answer.

(a) i) dy/dx= 2x-2-12x -1/2 ii) d 2 y/dx 2 = 2+6x -3/2 (b) Substitute x=4 into dy/dx= 2x-2-12x -1/2 Show that dy/dx= 0 and state 'hence there is a stationary point' (c) Substitute x=4 into d 2 y/dx 2 = 2+6x ...
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Work out the cube root of (64x1000).

We are asked to calculate (64x1000) 1/3 . We can use the laws of indices to write that expression as 64 1/3 x1000 1/3 , which is easier to evaluate. To work out 64 1/3 , we either spot the answer as 4 or tak...
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