H1011 Quizzes

Quiz 7: Two variable optimisation of surfaces
Question 1 Questions
Find the equation of the tangent to the curve $f\left(x\right)=xsin2x$ at $x=\frac{\pi }{4}$.
 a) $y=\frac{\pi }{4}x$ b) $y=x$ c) $y=\frac{\pi }{4}x-1$ d) $y=x+1-\frac{\pi }{4}$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}{f}^{\prime }\left(x\right)& =& sin2x+xcos2x,& \text{}\\ {f}^{\prime }\left(\frac{\pi }{4}\right)& =& 1,& \text{}\\ f\left(\frac{\pi }{4}\right)& =& \frac{\pi }{4}.& \text{}\end{array}$ So with $y=x+b,\frac{\pi }{4}=\frac{\pi }{4}+b,b=0.$
therefore the equation of the tangent is $y=x$.
Choice (c) is incorrect
Choice (d) is incorrect
Find two numbers whose difference is 20 and whose product is minimal.
 a) 1, 21 b) 19, -1 c) 10, -10 d) None of these.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Let the two numbers be $a,b$ where $a>b$.
Then $a-b=20$. Let the product of the numbers be $P$.
This is clearly a minimum value for the product because $\frac{{d}^{2}P}{d{a}^{2}}=2>0$ for all $a$.
Choice (d) is incorrect
Imagine constructing a closed steel box with volume 576 cm${3}^{}$ and with its base twice as long as it is wide. The steel costs \$40 per square metre. Determine the dimensions of the box that will minimise the cost of construction.
 a) 12 cm $×$ 6 cm $×$ 8 cm b) $12\sqrt{3}$ cm $×\phantom{\rule{1em}{0ex}}6\sqrt{3}$ cm $×\phantom{\rule{1em}{0ex}}\frac{8}{3}$ cm c) $12\sqrt[3]{2}$ cm $×\phantom{\rule{1em}{0ex}}6\sqrt[3]{2}$ cm $×\phantom{\rule{1em}{0ex}}4\sqrt[3]{2}$ cm d) None of the above

Choice (a) is correct!
The volume of the box is $2{x}^{2}h=576$, so $h=\frac{288}{{x}^{2}}$. The surface area of the box is $=S=4{x}^{2}+\frac{1728}{x}$. Therefore $\frac{dS}{dx}=8x-\frac{1728}{{x}^{2}}$. Hence, $\frac{dS}{dx}=0$, when $8{x}^{3}=1728$; that is, when $x=6$. Consequently, the dimensions of the box are 12 cm $×$ 6 cm $×$ 8 cm. Note that $S$ has a minimum at $x=6$ since $\frac{{d}^{2}S}{d{x}^{2}}=8+\frac{3456}{{x}^{3}}>0$ at $x=6$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate $f\left(x,y\right)={e}^{3x}cosy$ at $\left(2,\frac{\pi }{4}\right)$.
 a) $f\left(2,\frac{\pi }{4}\right)={e}^{\frac{3\pi }{4}}cos2$ b) $f\left(2,\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}{e}^{6}$ c) $f\left(2,\frac{\pi }{4}\right)=\frac{2}{\sqrt{2}}{e}^{6}$ d) $f\left(2,\frac{\pi }{4}\right)={e}^{\frac{3\pi }{2}}$

Choice (a) is incorrect
Choice (b) is correct!
When $x=2$ and $y=\frac{\pi }{4}$, ${e}^{3x}cosy={e}^{6}cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}{e}^{6}$.
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate $f\left(x,y\right)=sin2xy+cos3xy$ at $\left(1,\frac{\pi }{6}\right)$.
 a) $f\left(1,\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$ b) $f\left(1,\frac{\pi }{6}\right)=\frac{1}{2}$ c) $f\left(1,\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}+1$ d) $f\left(1,\frac{\pi }{6}\right)=\frac{3}{2}$

Choice (a) is correct!
$f\left(1,\frac{\pi }{6}\right)=sin\frac{\pi }{3}+cos\frac{\pi }{2}=\frac{\sqrt{3}}{2}+0=\frac{\sqrt{3}}{2}.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which point below is on the surface $z={x}^{2}+xy-{\left(4-y\right)}^{2}$ ?
 a) $\left(1,1,27\right)$ b) $\left(1,1,-7\right)$ c) $\left(-1,2,5\right)$ d) $\left(-1,2,-37\right)$

Choice (a) is incorrect
Choice (b) is correct!
When $x=y=1$, $z=1+1-{\left(3\right)}^{2}=-7$.
Choice (c) is incorrect
Choice (d) is incorrect
Which point below is on the surface $z=2{x}^{2}y+x-\sqrt{x+y}$ ?
 a) A$\left(2,2,16\right)$ b) B$\left(-1,-1,-1\right)$ c) C$\left(2,2,14\right)$ d) D$\left(1,1,1\right)$

Choice (a) is correct!
When $x=y=2$, $z=16+2-\sqrt{4}=16$.
Choice (b) is incorrect
Note that $\left(-1,-1\right)$ is not in the domain of the surface.
Choice (c) is incorrect
Choice (d) is incorrect
Which response below most accurately describes the intersection of the surface
$z=\sqrt{9-{x}^{2}-{y}^{2}}$
and the plane $z=2$ ?
 a) A circle of radius 5 in the plane $z=2$. b) A circle of radius 5 in the $xy$ plane. c) A circle of radius $\sqrt{5}$ in the plane $z=2$. d) A circle of radius 5 in the $xy$ plane.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The curve of intersection of the surface and the plane is given by $\sqrt{9-{x}^{2}-{y}^{2}}=2$. i.e ${x}^{2}+{y}^{2}=5$. This is a circle of radius $\sqrt{5}$ in the plane $z=2$.
Choice (d) is incorrect
Which response below most accurately describes the intersection of the surface $z=\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}$ and the plane $y=4$ ?
 a) An ellipse, $\frac{{x}^{2}}{36}-\frac{{y}^{2}}{16}=1$, in the plane $z=4$. b) A circle radius 2, ${x}^{2}+{y}^{2}=4$, in the plane $z=4$. c) A parabola $z=\frac{{x}^{2}}{9}-1$, in the plane $y=4$. d) A parabola $z=\frac{{x}^{2}}{9}-4$, in the plane $y=4$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The curve of intersection of the surface and the plane is given by $z=\frac{{x}^{2}}{9}-\frac{16}{4}$. i.e $z=\frac{{x}^{2}}{9}-4$ which is a parabola in the plane $y=4$.
Which response below most accurately describes the intersection of the surface $z=2{x}^{2}-2x-{y}^{2}-2y+3$ and the plane $x-y=0$?
 a) The parabola $z=3{y}^{2}+3$ in the plane $x-y=0$. b) The parabola $z=3{x}^{2}+3$ in the plane $x-y=0$. c) The parabola $z={x}^{2}-4x+3$ in the plane $x-y=0$. d) The intersection cannot be determined.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$x-y=0⇒y=x$ so $z=2{x}^{2}-2x-{x}^{2}-2x+3={x}^{2}-4x+3$.
Choice (d) is incorrect