### Math 20a Midterm 2 Review Problems

The Midterm will cover the material we did through Monday, November 10: Chapter 11.1-11.4, Chapter 13.1-13.7, Chapter 14.1-14.4 (only part of 14.3), Chapter 15.1-15.6. The test is cumulative, i.e. you are responsible for everything done since the beginning of the class, but the emphasis willl be on material covered since the last midterm. Each chapter has quite a few review problems; I have suggested some to work on. As for the regular problems, there are solutions in the book to the odd-numbered ones; I will post solutions to the even ones early next week.You might find the "concept check" sections of each chapter review, as well as true-false quizzes, to be helpful.

Chapter 15: 2,4,5,7,8,9,10,16,17,22,24,28,29,31,33,36,42,44 (`when' means that you are at some point *(x*_{0},y_{0}) and looking for the unit vector such that the directional derivative satisfies the given condition.), 46, 47.

**A few additional problems:**

A. We learned in chapter 11 that the slope of a parameterized curve* α (t) = (x(t),y(t))* at the point *(x(t*_{0}),y*(t*_{0}))* *is given by *y'(t*_{0})/*x'(t*_{0}). On the other hand, in section 15.5, we found out that the slope of the tangent at *(x*_{0}, y_{0}) to a curve in the plane, given by an equation *F(x,y) = 0, *is given by -F_{x}/F_{y}. Suppose that the curve with *F(x,y) = 0 *is parameterized by * α (t). *Explain why the two formulas for the slope give the same answer. (Hints: chain rule, implicit differentiation.)

B. Find the linearization of * f(x,y) =a*_{0} + ax + by + cx^{2} + dxy + ey^{2} at *(0,0). * Guess an extension of what you found to higher degree polynomials.

C. Let z = *F(x,y) = y*^{3}sin(xy) + e^{y}. Find a function* z=g(x) *which satisfies *g(3) = F(3,2), g'(3)= F*_{x}(3,2), g''(3) = F_{xx}(3,2), etc.* *

D. Consider a surface in R^{3} described in spherical coordinates by an equation of the form ρ=f(φ). What can you say about the shape of this surface? (Hint: there's no θ in the equation.) What will be its equation in cylindrical coordinates?

E. Can you have a function *f(x,y)* whose directional derivative *D*_{u}*f(x*_{0}, y_{0}) (at some point *(x*_{0}, y_{0}) ) is negative for every choice of unit vector **u**? Give an example, or explain why there is no such function.

F. Take the contour map that I handed out in class, and draw a few gradient vectors for the height function represented by the contours. Find as many places as you can where the gradient is the 0-vector. Try to figure out what's going on near one of the saddles.

G. Consider the trace of the graph of *z = f(x,y)* in the plane *y=k*. As we discussed in class,* f*_{xx} gives the concavity of this curve, because it is the second derivative of the curve defining this trace. Suppose that *f*_{xx}(x,k) > 0, and that the *3*^{d} derivative *f*_{xxy}(x,k) is positive. What does this say about the shape of the surface?

H. Contour maps don't tell you what goes on in between the contour lines. Illustrate this by finding two different functions *f(x,y)* and *g(x,y)* whose contour lines *f(x,y) = k *and* g(x,y)=k* are the same for *k=1,2,3,4,* etc. Do this either by sketching a picture, or by giving formulas for *f* and *g*.

I. Show that Clairaut's theorem (*F*_{xy }= *F*_{yx}) is true for polynomials of 2 variables, ie expressions like *F(x,y) =* *x*^{5} + 3x^{2}y^{7 }- 12xy^{3}. (Hint: it suffices to work with monomials, ie terms of the form *x*^{m}y^{n}.)