Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: in particular, the last few exercises may be very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, or are independently marked.Logarithms1. § Differentiate.(a) ln5√x(b)1 + ln t1 − ln t(c) 1/ ln(x)(d) lnra2− x2a2+ x2(e)ln(1 + ex) 2(f) log2(e−xcos πx)2. Compute the derivative of ln(xn) in two different ways: by using logarithm identities, and byusing the chain rule.3. § Find y0if xy= yx.4. Prove the product rule in the following manner: (a) Let f and g be two functions, and considerthe function lnf(x) g(x). Use logarithm rules to rewrite it as the sum of two functions. (b)Take derivatives of each side of your logarithm identity, using the chain rule. (c) Multiplyboth sides by the common denominator.5. § Compute y0by first simplifying ln y and then using the fact that (ln y)0= y0/y, where:y =4rx2+ 1x2− 1Word problems6. § If a stone is thrown vertically upward from the surface of the moon with a velocity of 10m/s, its heigh (in meters) after t seconds is h = 10t − .83t2.(a) What is the velocity of the stone after 3 seconds?(b) What is the velocity of the stone after it has risen 25 meters?7. § (a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing the solutionof water and NaClO3to evaporate slowly. If V is the volume of such a cube with sidelength x, calculate dV/dx when x = 3 mm, and explain its meaning.(b) Show that the rate of change of the volume of such a cube with respect to its edge length(i.e. dV/dx) is equal to half the surface area of the cube. Explain geometrically why thisis true.18. § Boyle’s Law states that when a sample of gas is compressed at constant temperature, theproduct of the pressure and the volume remains the same: P V = C.(a) Find the rate of change of volume with respect to pressure.(b) A sample of gas is in a container at low pressure and is compressed steadily (constantrate of increase in pressure) at constant temperature for 10 minutes. Is the volumedecreasing more rapidly at the beginning or the end of the 10 minutes?9. § Assume that in a particular reaction, one molecule of the product C is formed from onemolecule of the reactant A and one molecule of the reactant B, and that the initial concen-trations of A and B have the common value [A] = [B] = a moles per liter. Let x be theconcentration of molecule C. Thenx =a2ktakt + 1for some constant k.(a) Explain how x relates to the concentrations of molecules A and B at time t.(b) Find the rate of reaction at time t.(c) Show that x satisfies the differential equationdxdt= k(a − x)2Show that the right-hand side is proportional to the product of the concentrations attime t of molecules A and B.(d) What happens to the concentrations of A, B, and C as t → ∞?(e) What happens to the rate of reaction as t → ∞?Questions on earlier material10. What’s the derivative of sin2x? What’s the derivative of cos2x? What happens when youadd them together and why?11. Find numbers A and B so thatddx[Aexcos x + Bexsin x] = excos x12. Find numbers α and β so that y = eαxsin(βx) is a solution to the differential equation:y00+ 4y0+ 5y = 0Check that y = eαxcos(βx) is also a solution.13. § Show that every curve in the family y = ax3is orthogonal to every curve in the familyx2+ 3y2= b, where a and b range over all real
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