1.4. Homework Problems1. Graph the function t^{2} in the neighborhood of t=3, draw a tangent line, and use its slope to verify that the derivative equals 2t at this point. 2. Graph the function sin e^{t} in the neighborhood of t=0, draw a tangent line, and use its slope to estimate the derivative. Answer: 0.5403023058. (You will of course not get an answer this precise using this technique.) 3. Differentiate the following functions with respect to t: 1, 7, t, 7t, t^{2}, 7t^{2}, t^{3}, 7t^{3}. 4. Differentiate 3t^{7}4t^{2}+6 with respect to t. 5. Differentiate at^{2}+bt+c with respect to t. [Thompson, 1919] 6. Find two different functions whose derivatives are the constant 3, and give a geometrical interpretation. 7. Find a function x whose derivative is \(\dot{x}=t^7\). In other words, integrate the given function. 8. Find a function x whose derivative is \(\dot{x}=3t^7\). In other words, integrate the given function. 9. Find a function x whose derivative is \(\dot{x}=3t^74t^2+6\). In other words, integrate the given function. 10. Let t be the time that has elapsed since the Big Bang. In that time, one would imagine that light, traveling at speed c, has been able to travel a maximum distance ct. (In fact the distance is several times more than this, because according to Einstein's theory of general relativity, space itself has been expanding while the ray of light was in transit.) The portion of the universe that we can observe would then be a sphere of radius ct, with volume v=(4/3)π r^{3}=(4/3)π(ct)^{3}. Compute the rate \(\dot{v}\) at which the volume of the observable universe is increasing, and check that your answer has the right units, as in example 3 on Section 1.2. 11. Kinetic energy is a measure of an object's quantity of motion; when you buy gasoline, the energy you're paying for will be converted into the car's kinetic energy (actually only some of it, since the engine isn't perfectly efficient). The kinetic energy of an object with mass m and velocity v is given by K=(1/2)mv^{2}. For a car accelerating at a steady rate, with v=at, find the rate \(\dot{K}\) at which the engine is required to put out kinetic energy. \(\dot{K}\), with units of energy over time, is known as the power. Check that your answer has the right units, as in example 3 on Section 1.2. 12. A metal square expands and contracts with temperature, the lengths of its sides varying according to the equation ℓ=(1+α T)ℓ_{o}. Find the rate of change of its surface area a with respect to temperature. That is, find \(\dot{a}\), where the variable with respect to which you're differentiating is the temperature, T. Check that your answer has the right units, as in example 3 on Section 1.2. 13. Find the second derivative of 2t^{3}t. 14. Locate any points of inflection of the function t^{3}+t^{2}. Verify by graphing that the concavity of the function reverses itself at this point. 15. Let's see if the rule that the derivative of t^{k} is kt^{k1} also works for k<0. Use a graph to test one particular case, choosing one particular negative value of k, and one particular value of t. If it works, what does that tell you about the rule? If it doesn't work? 16. Two atoms will interact via electrical forces between their protons and electrons. To put them at a distance r from one another (measured from nucleus to nucleus), a certain amount of energy E is required, and the minimum energy occurs when the atoms are in equilibrium, forming a molecule. Often a fairly good approximation to the energy is the LennardJones expression \[ E(r) = k\left[\left(\frac{a}{r}\right)^{12}2\left(\frac{a}{r}\right)^6\right] , \]
where k and a are constants. Note that, as proved in chapter 2, the rule that the derivative of t^{k} is kt^{k1} also works for k<0. Show that there is an equilibrium at r=a. Verify (either by graphing or by testing the second derivative) that this is a minimum, not a maximum or a point of inflection. 17. Prove that the total number of maxima and minima possessed by a thirdorder polynomial is at most two. 18. Functions f and g are defined on the whole real line, and are differentiable everywhere. Let s = f+g be their sum. In what ways, if any, are the extrema of f, g, and s related? 19. Euclid proved that the volume of a pyramid equals (1/3)bh, where b is the area of its base, and h its height. A pyramidal tent without tentpoles is erected by blowing air into it under pressure. The area of the base is easy to measure accurately, because the base is nailed down, but the height fluctuates somewhat and is hard to measure accurately. If the amount of uncertainty in the measured height is plus or minus e_{h}, find the amount of possible error e_{V} in the volume. 20. A hobbyist is going to measure the height to which her model rocket rises at the peak of its trajectory. She plans to take a digital photo from far away and then do trigonometry to determine the height, given the baseline from the launchpad to the camera and the angular height of the rocket as determined from analysis of the photo. Comment on the error incurred by the inability to snap the photo at exactly the right moment. 21. Prove, as claimed on p. 10, that if the sum 1^{2}+2^{2}+…+n^{2} is a polynomial, it must be of third order, and the coefficient of the n^{3} term must be 1/3.
[Solutions of these problems are in the pdf version of the book]
