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With questions about projectiles, such as a ball travelling in the shape of a parabola through the air as in the above question, it is always good to split the problem into the horizontal and vertical com...

f'(x)=( sqrt (x^2+1) - x * ( x / sqrt (x^2 +1) ) ) / (x^2+1) = (x^2 + 1 + x^2) / ( (x^2 + 1) * sqrt ( x^2 + 1) ) =  1 / ( (x^2 + 1) * sqrt (x^2 + 1) ). 

f'(x) > 0 for any x => f is increasing...

for part a) let y=x^cos(X) , the ln(y)=ln(x^cos(X))=cos(x)ln(x), thus d/dx (ln(y(x)) = d/dx (cos(x)ln(x)), 1/y*dy/dx=cox(x)/x - sinxlnx => solve for dy/dx =>...

The point of intersection is where the lines would cross if we drew them on the same graph, in your exam you may be asked to find the coordinates of this point.

To do this we would first draw a ske...

Integrate ln(x) by parts with u=ln(x) and dv/dx=1 Gives    x+xln(x)-x+c Therefore xln(x)+c