Power Rule

A curve is defined by the equation $y=\frac{2x^2-1-2x\ln x}{x}$ , where $x>0$.

The curve has a point of inflection at the point P.

Find the equation of the tangent to the curve at the point P.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Find the $x$-value(s) of any stationary points on the graph of the function $f(x)=\frac{x^2+1}{x}$.

You must use calculus and show any derivatives that you need to find when solving this problem.

The graph of the function $y=4\sqrt{x}-x+2$, where $x>0$, has a stationary point at point Q.

Find the coordinates of point Q.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

For what value(s) of $x$ does the tangent to the graph of the function

$f(x)=2x-2\sqrt{x},x>0$, have a gradient of 1?

You must use calculus and show any derivatives that you need to find when solving this
problem.

The graph of the function $y=\frac{1}{x-3}+x$, $x\ne 3$, has two stationary points.

Find the $x$-coordinates of the stationary points, and determine whether they are local maxima
or local minima.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

The velocity of an object is modelled by the function

$v=2e^t+8e^{-t}$, for $t\ge 0$

where $v$ is the velocity of the object, in m s${}^{-1}$
and $t$ is the time in seconds since the start of the object’s motion.

Find the time when the acceleration of the object is 0.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Find the gradient of the curve $y=\frac{1}{x}-\frac{1}{x^2}$ at the point $\left(2,\frac{1}{4}\right)$.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Differentiate $y=1+x-\frac{1}{x}+\frac{1}{x^2}$.

The equation of motion of a particle is given by the differential equation

$\frac{d^{2}x}{d{t}^2}=-k^{2}x$

where $x$ is the displacement of the particle from the origin at time $t$, and $k$ is a positive
constant.

(i) Show that $x=A\cos kt+B\sin kt$, where $A$ and $B$ are constants, is a solution of the
equation of motion.

(ii) The particle was initially at the origin and moving with velocity $2k$.

Find the values of $A$ and $B$ in the solution $x=A\cos kt+B\sin kt$.

Find the gradient of the normal to the curve $y=x-\frac{16}{x}$ at the point where $x=4$.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Find the value(s) of $x$ for which the graph of the function $y=x+\frac{32}{x^2}$ has stationary points.

*Show any derivatives that you need to find when solving this problem.*

For what value of $k$ does the function $f(x)=x-e^x-\frac{k}{x}$ have a stationary point at $x=-1$?

Show any derivatives that you need to find when solving this problem.