The following is a graph of some function y=f(x).

(i) In your answer booklet graph y=f'(x) and y=f''(x) if at x=1 there is a point of inflexion.

(ii) Explain what happens to the curve at a point of inflexion.

$Find a primitive function of 8x-x^{-2}$

Graph the curves then shade the intersection of the regions defined by :

$2y\ge x^2-5 and y<x-1$

if  $f\left(x\right)=x+\frac{1}{x}$

Solve f (x) 2 = −2 . 

Show whether the function is odd, even or neither. 

Write down the domain and range of f (x ).

Solve $m^4+12m^2-64=0$

The line $3\mathcal{x}+5\mathcal{y}=k$ is a tangent to $\mathcal{y}={\mathcal{x}}^2-\mathcal{x}-1$

i) Explain why the discriminant of $3\mathcal{x}+5({\mathcal{x}}^2-\mathcal{x}-1)=k$ must equal zero.

ii) Hence, find the value of k.

$Determine whether the function𝑓(𝑥)=\frac{{\mathcal{x}}^2+4}{{\mathcal{x}}^3-\mathcal{x}} is odd, even, or neither$

Find the primitive function of $3-\frac{3}{{\mathcal{x}}^2}$

Use Simpson’s rule with 5 function values to find the approximate volume, to 2 decimal places, when the area bounded by the curve

$y=\frac{1}{\sqrt{4+x^2}}$

x-axis and the lines x =1 and x = 5, is rotated about the x-axis

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