Application of Derivatives — Practice

Question 1

Find the equation of the normal to $y=x^{3}-3x$ at the point where $x=2$.

Accepted Answer

At $x=2$, $y=2$ and $\dfrac{dy}{dx}=3x^{2}-3=9$.

Normal slope $=-	frac19$.

$y-2=-	frac19(x-2)$, that is $x+9y=20$.

Question 2

Show that the curves $y=x^{3}$ and $6y=7-x^{2}$ intersect at right angles at $(1,1)$.

Accepted Answer

First: $\dfrac{dy}{dx}=3x^{2}=3$ at $(1,1)$.

Second: $6\dfrac{dy}{dx}=-2x$, so $\dfrac{dy}{dx}=-	frac13$ at $(1,1)$.

Product $=-1$, so the curves are orthogonal. Both pass through $(1,1)$, since $1=1$ and $6=7-1$.

Question 3

Find the values of $a$ for which $f(x)=x^{3}+ax^{2}+3x+7$ is increasing on $\mathbb{R}$.

Accepted Answer

$f'(x)=3x^{2}+2ax+3\ge0$ for all $x$ needs $D\le0$.

$4a^{2}-36\le0$, so $a^{2}\le9$.

$-3\le a\le3$, endpoints included since a single touching zero is allowed.

Question 4

Find the greatest and least values of $f(x)=2x^{3}-15x^{2}+36x$ on $[1,5]$.

Accepted Answer

$f'(x)=6x^{2}-30x+36=6(x-2)(x-3)$, so stationary at $x=2,3$.

$f(1)=23$, $f(2)=28$, $f(3)=27$, $f(5)=55$.

Greatest $55$ at the right endpoint; least $23$ at the left endpoint. Neither extreme is at a stationary point.

Question 5

Classify the critical point of $f(x)=x^{5}$ at the origin, and of $g(x)=x^{6}$.

Accepted Answer

$f'(x)=5x^{4}\ge0$ with no sign change, so the origin is a stationary point of inflection and not an extremum.

$g'(x)=6x^{5}$ changes from negative to positive, so the origin is a minimum.

The second derivative test is silent for both, and only the sign of the first derivative separates them.

Question 6

A window is a rectangle surmounted by a semicircle, with total perimeter $10$ m. Find the dimensions admitting the most light.

Accepted Answer

With width $2r$ and height $h$: perimeter $2r+2h+\pi r=10$, area $A=2rh+	frac{\pi r^{2}}{2}$.

Eliminating $h$: $A=10r-2r^{2}-	frac{\pi r^{2}}{2}$.

$\dfrac{dA}{dr}=10-4r-\pi r=0$ gives $r=\dfrac{10}{4+\pi}$, and $\dfrac{d^{2}A}{dr^{2}}<0$ confirms a maximum.

Then $h=r$, so the rectangle is twice as wide as it is tall.

Question 7

Find the point on $y^{2}=4x$ nearest to $(2,-8)$.

Accepted Answer

Use $\left(t^{2},2tight)$. Then $D=\left(t^{2}-2ight)^{2}+(2t+8)^{2}$.

$\dfrac{dD}{dt}=4t\left(t^{2}-2ight)+4(2t+8)=4\left(t^{3}+16ight)$... expanding: $4t^{3}-8t+8t+32=4t^{3}+32$.

Setting this to zero gives $t^{3}=-8$, so $t=-2$ and the point is $(4,-4)$.

Question 8

Find the asymptotes of $y=\dfrac{x^{2}+1}{x-2}$ and the coordinates of its turning points.

Accepted Answer

$y=x+2+\dfrac{5}{x-2}$, so the asymptotes are $x=2$ and $y=x+2$.

$\dfrac{dy}{dx}=1-\dfrac{5}{(x-2)^{2}}=0$ gives $(x-2)^{2}=5$, so $x=2\pm\sqrt5$.

The corresponding values are $y=4\pm2\sqrt5$, a local maximum on the left branch and a local minimum on the right.

Question 9

Prove that $\ln(1+x)<x$ for every $x>0$.

Accepted Answer

Let $g(x)=x-\ln(1+x)$, so $g(0)=0$.

$g'(x)=1-\dfrac{1}{1+x}=\dfrac{x}{1+x}>0$ for $x>0$.

So $g$ is strictly increasing from zero and hence positive.

Question 10

For what values of $k$ does $x^{3}-6x^{2}+9x+k=0$ have three distinct real roots?

Accepted Answer

$f'(x)=3x^{2}-12x+9=3(x-1)(x-3)$, so critical points at $x=1$ and $x=3$.

$f(1)=4+k$ (local maximum), $f(3)=k$ (local minimum).

Three distinct roots need $4+k>0$ and $k<0$, that is $-4<k<0$.

Question 11

Water is poured into an inverted cone of semi-vertical angle $45^{\circ}$ at $2\ \mathrm{cm^{3}/s}$. How fast is the depth rising when the depth is $4$ cm?

Accepted Answer

With $r=h$, $V=	frac13\pi h^{3}$.

$\dfrac{dV}{dt}=\pi h^{2}\dfrac{dh}{dt}$, so $\dfrac{dh}{dt}=\dfrac{2}{16\pi}=\dfrac{1}{8\pi}$ cm/s.

The rate falls as the cone widens, which is the expected behaviour.

Question 12

Use a differential to approximate $\sqrt{50}$, and estimate the error.

Accepted Answer

$f(x)=\sqrt x$ at $x=49$: $f(49)=7$, $f'(49)=	frac{1}{14}$.

$\sqrt{50}\approx7+	frac{1}{14}\approx7.0714$.

The true value is $7.07107$, so the error is about $4	imes10^{-4}$, consistent with $	frac12\left|f''ight|\Delta x^{2}=	frac{1}{4\cdot343}$.

Practice — Application of Derivatives

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