Homework 8: Calculus and Analytic Geometry

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 11 February 2010

Problem A. Tangents and normals.

1. Find the equations of the tangent and the normal to the curve $y=x^4-6x^3+13x^2-10x+5$ at the point where $x=1.$
2. Find the equations of the tangent and the normal to the curve $y={\cot }^{2}x-2\cot x+2$ at the point where $x=1.$ at $x=\frac{\pi }{4}$
3. Find the equations to the tangent to the curve given by the equations $x=\theta +\sin \theta$ and $y=1+\cos \theta$ at $\theta =\frac{\pi }{4}$
4. Find the equations to the tangent to the curve given by the equations $x=a\cos \theta$ and $y=b\sin \theta$ at $\theta =\frac{\pi }{4}$
5. For a general $t$ find the equation of the tangent and normal to the curve given by the equations $x=a\cos t$ and $y=b\sin t$
6. For a general $t$ find the equation of the tangent and normal to the curve $x=a\sec t$ and $y=b\tan t$
7. For a general $t$ find the equation of the tangent and normal to the curve given by the equations $x=a\left(t+\sin t\right)$ and $y=b\left(1-\cos t\right)$
8. Find the equations of the tangent and normal to the curve $16x^2+9y^2=144$ at $\left(x_1,y_1\right)$ where $x_1=2$ and $y_1>0$
9. Find the equation of the tangent to the curve $y={\sec }^{4}x-{\tan }^{4}x$ at $x=\frac{\pi }{3}$
10. Find the equation of the normal to the curve $y={\left(\sin \left(2x\right)+\cot x+2\right)}^2$ at $\frac{\pi }{2}$
11. Find the equation of the normal to the curve $y=\frac{1+\sin x}{\cos x}$ at $x=\frac{\pi }{4}$
12. Show that the tangents to the curve $y=2x^3-3$ at the points where $x=2$ and $x=-2$ are parallel
13. Show that the tangents to the curve $y=x^2-5x+6$ at the points $\left(2,0\right)$ and $30$ are at right angles
14. Find the points on the curve $2a^{2}y=x^3-3a{x}^2$ where the tangent is parallel to the $x$-axis
15. For the curve $y\left(x-2\right)\left(x-3\right)=x-7$ show that the tangent is parallel to the $x$-axis at the points for which $x=7\pm 2{\left(5\right)}^{\frac{1}{2}}$
16. Find the points on the curve $y=4x^3-2x^5$ for which the tangent passes through the origin
17. Find the points of the circle $x^2+y^2=13$ for which the tangent is parallel to the line $2x+3y=7$
18. Find the point on the curve $y=3x^2+4$ at which the tangent is perpendicular to the line whose slope is $-\frac{1}{6}$
19. Find the equations of the normal to the curve $2x^2-y^2=14$ parallel to the line $x+3y=4$
20. Find the equation of the tangent to the curve $x^2+2y=8$ which is perpendicular to the line $x-2y+1=0$
21. Bonus problem If the straight line $x\cos \alpha +\mathrm{y}\sin \alpha =p$ touches the curve $${\left(\frac{x}{a}\right)}^{\frac{n}{n-1}}+{\left(\frac{y}{b}\right)}^{\frac{n}{n-1}}=1$$ show that ${\left(a\cos \alpha \right)}^n+{\left(b\sin \alpha \right)}^n=p^n$

Problem B. Optimization.

1. Find the local maxima and minima of $f\left(x\right)={\left(5x-1\right)}^2+4$ without using derivatives
2. Find the local maxima and minima of $f\left(x\right)=-{\left(x-3\right)}^2+9$ without using derivatives
3. Find the local maxima and minima of $f\left(x\right)=-\left|x-4\right|+6$ without using derivatives
4. Find the local maxima and minima of $f\left(x\right)=\sin 2x+5$ without using derivatives
5. Find the local maxima and minima of $f\left(x\right)=\left|\sin 4x+3\right|$ without using derivatives
6. Find the local maxima and minima of $f\left(x\right)=x^4-62x^2+120x+9$
7. Find the local maxima and minima of $f\left(x\right)=\left(x-1\right){\left(x+2\right)}^2$
8. Find the local maxima and minima of $f\left(x\right)=-{\left(x-1\right)}^3{\left(x+1\right)}^2$
9. Find the local maxima and minima of $f\left(x\right)=\frac{x}{2}+\frac{2}{x}$ for $x>0$
10. Find the local maxima and minima of $f\left(x\right)=2x^3-24x+107$ in the interval $\left\{1,3\right\}$
11. Find the local maxima and minima of $f\left(x\right)=\sin x+\frac{1}{2}\cos x$ in $0\le x\le \frac{\pi }{2}$
12. Show that the maximum value of ${\frac{1}{x}}^x$ is $e^{\frac{1}{e}}$
13. Show that $f\left(x\right)=x+\frac{1}{x}$ has a local maximum and a local minimum but the maximum value is less than the minimum value
14. Find the maximum profit a company can make if the profit function is given by $p\left(x\right)=41+24x-18x^2$
15. An enemy jet is flying along the curve $y=x^2+2$. A soldier is placed at the point (3,2). At what point will the jet be when the soldier and the jet are closest?
16. Find the local maxima and minima of $f\left(x\right)=-x+2\sin x$ in $\left[0,2\pi \right]$
17. Divide 15 into two parts so that the square of one times the cube of the other is maximum
18. Suppose the sum of two numbers is fixed. Show that their product is maximum exactly when one of them is half of the total sum
19. Divide $a$ into two parts so that the $p$th power of one times the $q$th power of the other is a maximum
20. What fraction exceeds its $p$th power by a maximum amount?
21. Find the dimensions of the rectangle of area 96 ${\mathrm{cm}}^2$ which has minimum perimeter. What is this minimum?
22. Show that the right circular cone with a given volume and minimum curved surface area has altitude equal to $2^{\frac{1}{2}}$ times the radius of the base
23. Show that the altitude of the right circular cone with a maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{4R}{3}$
24. Show that the height of a right circular cylinder with maximum volume that can be inscribed in a given right circular cone of height $h$ is $\frac{h}{3}$
25. A cylindrical can is made to hold 1 liter of oil. Find the dimensions of the can which will minimise the cost of the metal required to make the can.
26. An open box is to be made out of a given quntity of cardboard of area $p^2.$ Find the maximum volume of the box if its base is square.
27. Show that $f\left(x\right)=\sin x\left(1+\cos x\right)$ is maximum when $x=\frac{\pi }{3}$
28. An 8 inch oiece of wire is to be cut into two pieces. Figure out where to cut the wire in order to make the sum of the squares of the lengths of the two piece as small as possible
29. Find the dimensions of the maximum rectangular area that can be fenced with a fence 300 metres long
30. Given the perimeter of a rectangle show that its diagonal is minimum when it is a square. Make up a word problem for which this is a solution.
31. Prove that the rectangle of maximum area that can be inscribed in a given circle is a square. Make up a word problem for which this gives the solution
32. Show that the triangle of greatest area with given base and vertical angle is isoceles
33. Show that a right triangle with a given perimeter has greatest area when it is isoceles
34. Show that the angle of the cone wirth a given slant height and with maximum volume is ${\tan }^{-1}$ 2 1 2

Problem C. Related rates.

1. Find the rate of change of the volume of a sphere of radius $r$ with respect to a change in the radius
2. Find the rate of change of volume of a cylinder of radius $r$ and height $h$ with respect to a change in the radius
3. Find the rate of change of the curved surface of a cone of radius $rand\; height$ h$with\; respect\; to\; a\; change\; in\; the\; radius$
4. The side of a square is increasing at a rate of 0.2 cm/s. Find the rate of increase of the perimeter of the square
5. A balloon which always remains spherical is being inflated by pumping in 900 cubic centimeters of gas per second. Find that rate at which the radius of the balloon is increasing when the radius is 15cm.
6. The surface area of a spherical bubble is increasing at 2 $\frac{{\mathrm{cm}}^2}{s}$. When the radius of the bubble is 6 $\mathrm{cm}$ what rate is the volume of the bubble increasing?
7. The bottom of a rectangular swimming pool is 25 $\times$ 40 metres. Water is pumped into the tank at the rate of 500 cubic meters per minute. Find the rate at which the water level in trhe tank is rising.
8. A runner runs around a circular track of radius 100m at a constant speed of 7m/s. The runner's friend is standing 200m from the center of the track. How fast is the distance between the two changing when the distance between the two is 200m.
9. A streetlight is at the top of a 15 foot tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path
1. How fast is the tip of his shadow moving when he is 40m from the pole?
2. How fast is his shadow lengthening at that point?
10. A lighthouse is on a small island 3km away from the nearest point $P$ on a straight shoreline and its light turns 4 revolutions per minute. How fast is the beam of light moving along the shoreline if it is 1km from $P$?
11. A boat is pulled into the dock by a rope attached to the bow of the boat passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s then how fast is the boat approaching the dock when it is 8m from the dock?
12. Gravel is being dumped from a conveyer belt at a rate of 30 ${\mathrm{ft}}^3$/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high?
13. Water is dripping from a tiny hole in the vertex at the bottom of a conical funnel at a uniform rate of 4 cubic centimetres per second. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water. given that the vertical angle of the funnel is ${120}^{°}$
14. Water is leaking out of an inverted conical tank at a rate of $10000{\mathrm{cm}}^3/\min$ at the same time that the water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water height is rising at a rate of 20 cm/min when the height of the water is 2 m, fnd the rate at which water is being pumped into the tank.
15. Oil is leaking from a cylindrical drum at a rate of 16 millilitres per second. If the radius of the drum is 7 cm and its height is 60 cm find the rate at which the level of the oil is changing when the level of the oil is 18 cm.
16. A ladder 10 feet lnog rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft/s how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{\pi }{4}$ radians?
17. A ladder 13 metres long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at a rate of 2 m/s. How fast is its height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
18. A man is moving away from a 40 metre tower at a speed of 2 m/s. Find the rate at which the angle of elevation of the top of the tower is changing when he is at a distance of 30 metres from the foot of the tower. Assume that the eye level of the man is 1.6 metres from the ground.
19. Find the angle which increases twice as fast as its sine.
20. A television is positioned 4000 ft from the base of a rocket launching pad. A rocket rises vertically and its speed is 600 ft/s when it has risen 3000 feet.
1. How fast is this distance from the television camera to the rocket changing at that moment?
2. How fast is the camera's angle of elevation changing at that same moment?

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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