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Curve Sketching, Integration, Stationary Points and Asymptotes

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Coursework 2
Question 2

a) For the curve with the equation y = x^3 + 3x^2 - 2

i) Find the position and nature of any stationary points.
ii) Make up tables of signs for y, y' and y''. Use these to help you make a sketch of the curve.

b) Using your sketch of the curve y = x^3 + 3x^2 - 2

i) Shade in the region representing S x^3 + 3x^2 - 2 dx 1 ---> -3
ii) Without evaluating the integral, explain why its value is zero.
iii) Evaluate the integral to confirm your answer.

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Questions about curve sketching, integration, stationary points and asymptotes are answered in detail. The solution is comprehensive and well presented.

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Question 2

a) For the curve with the equation

i) Find the position and nature of any stationary points.

Stationary point is a point at which the derivative of the function vanishes

f''(x) = 3x2 + 6 x

Equating to zero, 3x2 + 6 x = 0  3x (x+2)=0  x = 0, -2

To determine the nature, let's find, f'' (x) = 6 x + 6

The specific nature of a stationary point at x can be determined by examining the second derivative f''(x):
• If f''(x) < 0, the stationary point at x is a maximal extremum.
• If f''(x) > 0, the ...

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