Math 5BI: Problem Set 1 Linearizing functions of several variables

Transcription

1 Math 5BI: Problem Set Linearizing functions of several variables March 9, A. Dot and cross products There are two special operations for vectors in R that are extremely useful, the dot and cross products. Definition. Suppose that v = (v, v, v ) and w = (w, w, w ) are two vectors in R. The dot product of v and w is the real number v w = v w + v w + v w. Problem.. Find the dot product of the vectors v = (,, 6) and w = (,, 5). The length of a vector v = (v, v, v ) R is defined to be v = v v = v + v + v. Problem.. Find the length of the vector v = (,, ). Note that v w is just the distance from v = (v, v, v ) to w = (w, w, w ) in R. Here is an important fact from trigonometry called the law of cosines. If a, b and c are lengths of the sides of a triangle, then c = a + b ab cos θ, where θ is the angle between the sides of length a and b. Problem.. a. Use the law of cosines to derive the formula where θ is the angle between v and w. v w = v w cos θ, b. Show that two nonzero vectors v = (v, v, v ) and w = (w, w, w ) are perpendicular if and only if v w =.

2 We can represent a point in R as a vector x = (x, y, z). Problem.4. a. Under what conditions is the vector perpendicular to the vector (5,, )? x (, 7, ) = (x, y, z) (, 7, ) b. Find an equation that represents the plane which contains the point p = (, 7, ) and is perpendicular to the vector (5,, ). c. Find an equation fhat represents the plane which contains the point p = (x, y, z ) and is perpendicular to the vector a = (a, a, a ). The standard basis for R is {i, j, k} where i = (,, ), j = (,, ), k = (,, ). Can you show that it actually is a basis for the vector space R? Definition. Suppose that v = (v, v, v ) and w = (w, w, w ) are two vectors in R. The cross product of v and w is the vector i j k v w = v v v w w w = v v w w i + v v w w j + v v w w k. Problem.5. a. Find the cross product v w of the vectors v = (,, 6) and w = (,, 5). b. Show that the result v w is perpendicular to v = (,, 6) and w = (,, 5). c. Show that if v = (v, v, v ) and w = (w, w, w ) are two linearly independent vectors in R, then v w is perpendicular to v and w. Problem.6. a. Find a vector that is perpendicular to the vectors (,, 6) and (,, 4). b. Find an equation of the plane which is tangent to the vectors (,, 6) and (,, 4) and contains the point (,, ). Problem.7. a. Show that if v = (v, v, v ) and w = (w, w, w ) are two vectors in R, then v w + v w = v w. b. Find a formula for v w in terms of v, w and the angle θ between v and w. We have described two very important operations on vectors in R, the dot and cross products. What are their properties? For example, are they commutative? Can you define a cross product for vectors in R 4? Why or why not?

3 B. Partial derivatives Recall that in first year calculus, the derivative determines the equation of the line tangent to the graph of a function. If f(x) is a differentiable function and x is a value for the variable x, then the line tangent to the graph of the curve y = f(x) is given by the equation ( ) df y = f(x ) + (x )(x x ). () ( ) In this equation x, f(x ), and df (x ) are constants, while x and y are variables. The function L(x) = f(x ) + ( ) df (x )(x x ) will be called the linearization of f(x) at x. (Strictly speaking, it is not usually a linear function, but an affine function, which is the same thing as a linear function plus a constant.) The tangent line to the graph of y = f(x) at (x, f(x )) is the graph of the linearization y = L(x) of f(x) at x. Problem.. Find the equation of the tangent line to the curve y = x + at the point (, 5). If z = f(x, y), where f is a smooth function and x and y are values for the variables x and y, then the linearization of f at (x, y ) is the affine function (linear function plus a constant) whose graph is the tangent plane to the graph of f at the point (x, y, f(x, y )). To find this linearization, we need to take partial derivatives. The partial derivative of f(x, y) with respect to x at (x, y ) is given by the formula x (x f(x + h, y ) f(x, y ), y ) = lim. h h Thus the partial derivative of f(x, y) with respect to x at (x, y ) is the ordinary derivative of the function x f(x, y ) at the point x. Similarly, the partial derivative of f(x, y) with respect to y at (x, y ) is y (x f(x, y + k) f(x, y ), y ) = lim. k k Definition. Let f(x, y) be a function of two variables. We say that f is continuously differentiable if it possesses partial derivatives x (x, y) and y (x, y) which vary continuously with the point (x, y).

4 Problem.. If f(x, y) = x + y, what is (/ x)(, )? To determine the linearization of a function f(x, y) at a given point (x, y ), we consider two parametrized curves which lie on the graph of f and intersect at the point (x, y, f(x, y )): γ : R R, γ (x) = (x, y, f(x, y )), γ : R R, γ (y) = (x, y, f(x, y)). The parameters on these two curves are x and y, respectively. vectors to these curves are v =, v = (/ x)(x, y ) (/ y)(x, y ) The velocity. () Problem.. a. Sketch the graph of z = f(x, y) = x + y. b. For this choice of f, find the velocity vector to the curve γ : R R, γ (x) = (x,, f(x, )) at the point x = x =. c. Find the velocity vector to the curve γ : R R, γ (y) = (, y, f(, y)) at the point y = y =. d. Find a vector perpendicular to the surface z = x + y at the point (,, ). e. Find the equation of the plane tangent to the surface z = x +y at the point (,, ). Problem.4. a. Suppose that f(x, y) is a continuously differentiable function of two variables. Find a vector n perpendicular to the surface z = f(x, y) at the point (x, y, f(x, y )). b. Find an equation for the plane tangent to the surface z = f(x, y) at the point (x, y, f(x, y )). c. Find the function L(x, y) = ax + by + c which most closely approximates f(x, y) near the point (x, y ). (We would call this the linearization of f at the point(x, y ).) It follows from the previous problem that the tangent plane to the surface z = f(x, y) is horizontal if and only if ( ) ( ) (x, y ) = (x, y ) =. () x y 4

5 A point satisfying () is called a critical point of the function f. If one thinks of the graph of f as representing a mountain range, then mountain peaks, mountain passes, and bottoms of lakes are critical points of f. In particular, if a function f(x, y) assumes a local maximum or a local minimum at a point (x, y ), then (x, y ) must be a critical point for f. This last fact can be used to find the candidates for maxima and minima of a given function of two variables. Problem.5. a. Suppose two lines are given in parametric form by x(t) = + t, y(u) = u. To find the distance between these two lines, we first let f(t, u) = (distance from the point x(t) to the point y(u)). Find the critical points for the function f(t, u). Problem.6. Let Π be the plane in R defined in parametric form by x(s, t) = + s + t. a. Find the critical points of the function f(s, t) = (distance from the point (, 5, ) to the point x(s, t)). b. Find the distance from the plane Π to the point (, 5, ). Homework. Due Friday, April,. H..a. Use partial derivatives as described in this section to find the linearization of the function z = f(x, y) = ( y.x)x at the point (x, y) = (,.9). b. Use partial derivatives to find the linearization of the function at the point (x, y) = (,.9). w = g(x, y) = ( + x)y c. Using the previous parts of this problem, find the linearization of the nonlinear system of differential equations at the point (x, y) = (,.9). /dt = f(x, y) = ( y.x)x, dy/dt = g(x, y) = ( + x)y. 5