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.Chain Rule1. § Differentiate.(a) (4x − x2)100(b) (1 + x4)3/2(c)3√1 + tan t(d)(y − 1)4(y2+ 2y)5(e) a3+ cos3x(f) (t4− 1)3(t3+ 1)4(g) 101−x2(h)eu− e−ueu+ e−u(i) tan2(3θ)(j) ek tan√x(k) sin(sin(sin x))(l)y2y + 122. § Suppose that f is differentiable on R. Let F (x) = f(ex) and G(x) = ef(x). Find expressionsfor F0(x) and G0(x).3. § For what values of r does the function y = erxsatisfy the differential equation y00+5y0−6y =0?4. § Find the 50th derivative of y = cos 2x. Find the 1000th derivative of f(x) = xe−x.5. § Air is being pumped into a spherical balloon. At any time t, the volume of the balloon isV (t) and the radius is r(t). What do the derivatives dV/dr and dV/dt represent? What isthe relationship between dV/dt, r, and dr/dt?6. § Use the chain rule to prove that the derivative of an even function is an odd function, andthat the derivative of an odd function is an even function.7. § If n is a positive integer, prove that:ddxsinnx cos nx= n sinn−1x cos(n + 1)xFind a similar formula forddxcosnx cos nx.8. The Leibniz notation makes the chain rule very natural looking:dydx=dydududx1However, the corresponding formula for the second derivative —d2ydx2=d2ydu2du2dx2=d2ydu2dudx2—is false. Instead, prove the following “chain rule for second derivatives”, by using the chainand product rules:d2ydx2=d2ydu2dudx2+dydud2udx29. (a) Let f be a differentiable function, such that f0(x) > 0 for all x. By drawing a graph,show that f is an increasing and hence one-to-one.(b) Let g be the inverse of f , so that f(g(x)) = g(f(x)) = x. Use the chain rule to find thederivative of g in terms of the derivative of f.(c) Show that ln0x — the derivative of ln x — is 1/x.Harder questions on earlier material10. Write out the first few derivatives (f, f0, f00, . . . ) of f(x) = xex. Do you notice a pattern?11. (a) Prove that if p is a polynomial of degree n, then the derivative of p(x) exis q(x) ex, whereq is also a polynomial of degree n.(b) Let f(x) = p(x) exwhere p is a polynomial. What is limx→−∞f(n)(x)?12. Let r be a rational function, so that r(x) = p(x)/q(x) for some polynomials p and q. Definethe degree of r to be deg p − deg q, where deg p is the degree of p, i.e. the highest power of anon-zero term in p. Prove that r0, the derivative of r, is a rational function, and prove thatdeg r0= deg r − 1.13. (a) Use the definition of derivative to prove the product rule.(b) Use the product rule to prove the quotient rule.(c) Let p be a polynomial. Use the product rule, but not the chain rule, to prove thatddxp(q(x)) = p0(q(x)) q0(x).In fact, from just the product rule, you can prove the chain rule provided that the outerfunction is a rational function (ratio of two polynomials).14. What’s the derivative of sin2x? What’s the derivative of cos2x? What happens when youadd them together and why?15. Find numbers A and B so thatddx[Aexcos x + Bexsin x] = excos x16. Let f(x) = ax. Then f0(0) = ln a. (Why?) Let’s say we didn’t know that. Define thefunction `(a) for a > 0 by `(a) =ddxaxx=0. Use the product rule to show directly that`(ab) = `(a) + `(b). Use the quotient rule to show directly that `(a/b) = `(a) − `(b). Showdirectly that `(1) =
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