Extension Exercise

L3CALC SCH_CALC

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10-a,b) a) Find values of $a$ and $b$ such that the curve

$$y=\frac{ax-b}{x^2-1}$$

has a turning point at $(3,1)$, and establish the nature of that turning point.

b) Sketch the curve.

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10-2a,b)

$$f(x)=x^{k+1}+k^k$$

where $k$ is an integer greater than 1.

a) Determine an expression, in terms of $k$, for the $x$-coordinates of any stationary point(s) on the graph of $f(x)$.

b) Determine an expression, in terms of $k$, for the $x$-coordinates of any inflection points on the graph of $f(x)$.

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10-3a,b)

a) Let $g(x)$ be a twice-differentiable function which satisfies this condition:

$g^{\prime }(x)=0$ implies $g^{\prime \prime }(x)\ne 0$ for all values of $x$.

If $h(x)=e^{g(x)}$, find $h^{\prime \prime }(x)$ and prove that $g(x)$ and $h(x)$ take local maximum values at exactly the same points.

b) Verify that if $g(x)=x+2\sin (x)$, then $g^{\prime }(x)=0$ implies $g^{\prime \prime }(x)\ne 0$. Hence (or otherwise) find the points in the interval $-2\pi <x<2\pi$ at which $h(x)=e^{x+2\sin (x)}$ takes local maximum values.

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10-4ai,aii,bi,bii) ai) On the same diagram, roughly sketch the diagrams of $y=\pi \sin (x)$ and $y=x$ for $0\le x\le \pi$. Use your diagram to deduce that there is a number $\alpha$ between $0$ and $\pi$ such that $\pi \sin (\alpha )=\alpha$.

aii) Show that:

$$\frac{2\pi }{3}<\alpha <\frac{3\pi }{4}$$

but do not try to evaluate $\alpha$.

bi) Let:

$$f(x)=\sqrt{{\pi }^2-x^2}\cos (x)-x\sin (x),-\pi \le x\le \pi$$

Prove that $f(x)$ is an even function, and evaluate $f(x)$ at $x=0,\frac{\pi }{3},\frac{\pi }{2},\alpha ,\pi$.

bii) Show that $f(x)$ has a turning point at $x=\alpha$. Find all the turning points of $f(x)$ for $-\pi \le x\le \pi$ and determine their nature.

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12-a,b,c,d) a) A tangent to the curve of $y=x^2-2x+3$ passes through the point $(4,7)$. Find the point(s) where the tangent touches the curve.

b) One of the points on the graph of $y=3x^2+4$ can be written as $(a,b)$, where $a>0$. The tangent at $(a,b)$ makes an angle of $45^o$ with each axis. Find where the tangent cuts the $x$-axis.

c) $f(x)=a{x}^2+bx+c$ has equal and real roots at points $P$ and $Q$, where $P=(p,0)$ and $Q=(q,0)$. Show that the normals to the parabola at $P$ and $Q$ intersect on the line:

$$x=\frac{p+q}{2}$$

d) A tangent to the curve $y=x^2+4$ passes through the point $(-2,7)$. Find the equation(s) of this tangent.

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13-1a,1b,1c,2a,2b,3) 1a) A curve has the equation:

$$x^3-3xy+y^3=0$$

Write down $\frac{dy}{dx}$ in terms of $x$ and $y$.

1b) Write down the coordinates of any points on the curve where $x=y$.

1c) There is one point on the curve where the gradient is 0. Locate this point, and determine whether it is a local maximum or a local minimum. (Note: the formula for $\frac{dy}{dx}$ is undefined when $y=x$).

2a) Use a graphics calculator or an appropriate graphics package to sketch the graph of $\sqrt{x}+\sqrt{y}=c$ for different values of $c$.

2b) Use implicit differentiation to help show that the sum of the intercepts made on the $x$ and $y$-axes by the tangent to $\sqrt{x}+\sqrt{y}=c$, at any point, is constant.

3) If $x=y\ln (xy)$, prove:

$$\frac{dy}{dx}=\frac{y(x-y)}{x(x+y)}$$