A curve $C$ with equation $y=-x^3+2x^2+8x$ crosses the $x$ axis at the points $A$, $O$, and $B$, where $O$ is the origin.
Using an appropriate algebraic method, find the coordinates of $A$ and $B$.
Sketch the graph of $y=-x^3+2^2+8x$.
Hence find the total area of the finite regions bounded by the curve and the $x$ axis.
$A(-2,0)$ and $B(4,0)$
negative cubic through $A$, $B$ and the origin
$\dfrac{148}{3}$
Q2
Answer
Prove that if $1+3x^2 + x^3 < (1+x)^3$, then $x>0$.
Show, by means of a counterexample, that $1 + 3x^2+x^3 < (1+x)^3$ is not true for all values of $x$.
Expand brackets and compare
e.g. $x = 0$
Q3
Answer
The curve with equation $y=f(x)$ passes through the point $(4,19)$. Given that $$f'(x) = 15x\sqrt{x}-\dfrac{40}{\sqrt{x}}$$ find $f(x)$.
$6x^{\frac{5}{2}} - 80x^{\frac{1}{2}} - 13$
Q4
Answer
Find all the solutions to the equation $8-7\cos x = 6\sin^2x$ in the interval $0 \leq x \leq 360^{\circ}$, giving your answers to 1 decimal place where appropriate.