Problem Set - Differentiation

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: 2 March 2010

Motion.

	What do distance, speed and acceleration have to do with calculus? Explain thoroughly.
	A particle, starting from a fixed point $P,$ moves in a straight line. Its distance from $P$ after $t$ seconds is $s=11+5t+t^3$ metres. Find the distance, veloity and acceleration of the particle after 4 seconds, and find the distance it travels during the 4th second.
	The displacement of a particle at time $t$ is given by $x=2t^3-5t^2+4t+3.$ Find the time when the acceleration is $8\mathrm{cm}/s^2$ and the velocity and displacement at that instant.
	A particle moves along a straight line so that after $t$ seconds its distance from a fixed point $P$ on the line is $s$ metres, where $s=t^3-4t^2+3t.$ Find when the particle is at $P$ and its velocity and acceleration at those times.
	A particle moves along a straight line according to the law $s=a{t}^2-2bt+c,$ where $a,b,c$ are constants. Prove that the acceleration of the particle is constant.
	If a particle moves along a straight line so that the distance described is proportional to the square of the time elapsed prove that the velocity is proportional to the time elapsed and and the rate of increase of the velocity is constant.
	A car starts from rest and moves a distance $s$ metres in $t$ seconds, where $s=a\cos t+b\sin t.$ Show that the acceleration at time $t$ is the negative of the distance travelled in $t$ seconds.
	A particle moves along a straight line according to the law $s=t^3-6t^2+19t-4.$ Find it displacement and acceleration when its veolicty is $7m/sand$ its displacement and velocity when its acceleration is $6m/s^2.$
	The distance $s$ in metres travelled by a particle in $t$ seconds is given by $s=a{e}^t+b{e}^{-t}.$ Show that the acceleration of the particle at time $t$ is equal to the distance that the particle travels in $t$ seconds.
	A particle moves in a line according to the law $s=a{t}^2+bt+c$ where $a,bc$ are constants and $s$ is the distance of the partice from a fixed point $P$ after $t$ seconds. Initially the particle is 10cm away from $P$ and its initial velocity is 12 cm/s. If the particle moves with uniform acceleration of 4 $\mathrm{cm}/s^2$ find the distance travelled by it during the 7th second.
	The displacement of a particle moving in a straight line is $x=2t^3-9t^2+12t+1$ metres at time $t.$ Find the velocity and acceleration at $t=1$ second the time when the particle stops momentarily, and the distance between two stops.
	The particle moves in a straight line according to the law $s=6t^3+20t^2+9t$ where $s$ is in centimetres and $t$ is in seconds. Find the initial velocity and acceleration of the particle.
	A particle is moving on a line according to the law $s={\tan }^{-1}t+a{t}^2+bt+c$ where $a,b,c$ are constants. Given that, at $t=1,$ $s$ is 3.5m, the velocity is 3 m/s, and the acceleration is 1.5 $m/s^2,$ find the values of $a,b,c.$
	The height of a stone thrown vertically upwards is given by $s=49t-4.9t^2$ where $x$ is in metres and $t$ is in seconds. Find its velocity at $t=1.$ At what time is its velocity 0? What is the maximum value of $s$?
	An arrow shot vertically upwards has equation of motion $s=ut-4.9t^2$ where $s$ is in metres and $t$ is in seconds. Find the time that it takes to reach a height of 117.6 metres. What is its velocity after 8 seconds. How long before it hits the ground?
	A ball projected vertically upwards has equation of motion $s=ut-4.9t^2$ where $s$ is in metres and $t$ is in seconds and $u$ is the initial velocity. If the maximum height reached by the ball is 44.1 metres find the value of $u.$
	A shot fired vertically upwards is known to be at a point $A$ at the end of 2 seconds and also there after 3 more seconds. The equation of motion of the bullet is $s=ut-4.9t^2$ where $s$ is in metres and $t$ is in seconds and $u$ is the initial velocity. Find the height of the point $A$ above the point where the shot is fired.
	A particle falls from the top of a tower and in the last second before it hits the ground falls 9/25 of the total height of the tower. Find the height of the tower.

Applications of the exponential function

	Find all functions $y\left(t\right)$ such that ${\displaystyle \frac{dy}{dt}}=ky$ and $y\left(a\right)=b,$ where $k,a$ and $b$ are constants.
	What is the idea behind comuting radioactive decay? Explain why exponential functions arise?.
	What is the idea for computing population growth? Explain why exponential functions are used.
	Explain why exponential functions are used to compute money owed on loans. Explain why the limit ${\displaystyle \underset{n\to \infty }{\lim }{\left(1+\frac{a}{n}\right)}^n}$ is used for computing interest.
	What is the ide for computing temperatures of objects during cooling? Explain why exponential functions appear.
	If you borrow $500 on a credit card at 14% interest find the amounts due at the end of 2 years in the interest is compounded (a) annually, (b) quarterly, (c) monthly, (d) daily, (e) hourly, (f) continuously.
	If you buy a $24,000 car at put 15% down and take out a 3 year loan at 7% per year compute how much your monthly payments are if the interest is compounded continuously.
	If you buy a $24,000 car at put 15% down and take out a 3 year loan at 7% per year compute how much your payment would be if you paid in one lump sum at the end of 3 years.
	If you buy a $24,000 car at put 15% down and take out a 3 year loan at 7% per year compute how much you pay in the first month.
	If you buy a $200,000 home and put 10% down and take out a 30 year fixed rate mortgage at 8% per year compute how much your monthly payments are if the interest is compounded continuously.
	If you buy a $200,000 home and put 10% down and take out a 30 year fixed rate mortgage at 8% per year compute how much your payment would be if you paid in one lump sum at the end of 30 years.
	If you buy a $200,000 home and put 10% down and take out a 30 year fixed rate mortgage at 8% per year compute how much you pay in the first month.
	A roast turkey is taken from an oven when its temperature has reached 185C and is placed on a table in a room where the temperature is 20C. Assume that it cools st a rate proportional to the difference between its current temperature and the room temperature. If the temperature is 150F after half an hour, what is the temperature after one hour? When will the turkey have cooled to 100C?
	Radiocarbon dating works on the principle that $C^{14}$ decays according to radioactive decay with a half life of 5730 years. A parchment fragment was discovered that had about 74% as much $C^{14}$ as does plant material on earth today. Estimate the age of the parchment.
	After 3 days a sample of radon-222 decayed to 58% of the original amount. What is the half life of radon-222? How long would it take for the sample to decay to 10% of its original amount?
	Polonium-210 has a half life of 140 days. If a sample has a mass of 200 mg find a formula for the mass that remains after $t$ days. Find the mass after 100 days. When will the mass be reduced to 10 mg? Sketch the graph of the mass function.
	If the bacteria in a culture increase continuously at a rate proportional to the number present and the initial number is $N_0$, find the number at time $t.$
	If an object at a rate proportional to the difference between its temperature and the temperature of its surroundings, the initial temperature of the object is $T_0$ and the temperature of the surroundings is a constant temperature $S$, what is the temperature of the object at time $t$?
	If a radioactive substance disintergrates at a rate proportional to the amount present how much of the substance remains at time $t$ if the initial amount is $Q_0$?
	Experts believe that the world's farms can feed about 10 billion people. The 1950 world population was 2517 million people and the 1992 world population was 5.4 billion people. When will we run out of food?
	Suppose that the GNP of a country is increasing at an annual rate of 4%. How many years, assuming that rate of growth continues, are required to double the present GNP?
	What percent of a sample of radium-226 remains after 100 years? The half life of radium-226 is 1620 years.
	A sample contains 4.6 mg of iodine-131. How many mg will remain after 3.0 days? The half life of iodine-131 is 8.0 days.
	To majority of naturally occuring rhenium is rhenium-187, which is radioactive and has a half life of $7\times {10}^{10}$ years. In how many years will 5% of the earth's rhenium-187 decompose?
	A piece of paper from the Dead Sea scrolls has found to have a carbon-14 to carbon-12 ratio 79.5% of that of a plant living today. The half life of carbon-14 is 5720 years. Estimate the age of the paper.
	The charcoal from ashes found in a grave gave a carbon-14 activity of 8.6 counts per gram per minute. Calculate the age of the charcoal (wood from a growing tree gives a count of 15.3). For carbon-14, the half life is 5720 years.
	In a certain activity meter, a pure sample of strontium-90 has an activity (rate of decay) of 1000.0 disintergrations per minute. If the activity of this sample after 2.00 years is 953.2 disintergrations per minte, what is the half life of strontium-90?
	A sample of wood from an artifact in an Egyptian tomb has 54.2% of the carbon-14 of a piece of freshly cut wood. In approximately what year was the old wood cut? The half life of carbon-14 is 5720 years.

Logarithmic differentiation.

	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\frac{{\left(x+2\right)}^{\frac{5}{2}}}{{\left(x+6\right)}^{\frac{1}{2}}{\left(x+3\right)}^{\frac{7}{2}}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(x+1\right)}^2{\left(x-2\right)}^3\left(x+4\right)\ln x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=\sqrt{\frac{\left(x-a\right)\left(x-b\right)}{\left(x-p\right)\left(x-q\right)}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\sin x\right)}^{\ln x}}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\sin x\right)}^{\cos x}}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\sin x\right)}^{\tan x}+{\left(\tan x\right)}^{\sin x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=y^x+x^y=a.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x+y=x^y}.$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\cos x\right)}^y={\left(\sin y\right)}^x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=a^x+e^{\tan x}+{\left(\cot x\right)}^{\cos x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\tan x\right)}^{\cot x}}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^x+x^{\frac{1}{x}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\sec x\right)}^{\csc x}+{\left(\csc x\right)}^{\sec x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\log }_{y}x.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\cos x\right)}^{{\left(\cos x\right)}^{{\left(\cos x\right)}^{{{.}^{.}}^{.}}}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^{x^{x^{{.}^{{.}^{.}}}}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^{y^x}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y=x^{\frac{1}{x}}.}$
	Find ${\displaystyle \frac{dy}{dx}}$ when ${\displaystyle y={\left(\frac{x^x+x^{-x}}{x^x-x^{-x}}\right)}^{\frac{1}{2}}..}$
	If $x^m{y}^n={\left(x+y\right)}^{m+n}$ show that ${\displaystyle \frac{dy}{dx}=\frac{y}{x}.}$

References

[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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