Exam 2 Review Sheet. r(t) = x(t), y(t), z(t)

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1 Exam 2 Review Sheet Joseph Breen Particle Motion Recall that a parametric curve given by: r(t) = x(t), y(t), z(t) can be interpreted as the position of a particle. Then the derivative represents the particle s velocity: v(t) = r (t) = x (t), y (t), z (t) and the second derivative represents the particle s acceleration: a(t) = v (t) = r (t) = x (t), y (t), z (t) The speed is given by the magnitude of the velocity vector: v(t). We can break the acceleration vector into two components: the component in the tangential direction of the curve, and the component in the normal direction of the curve: N(t) a(t) T(t) r(t) The formulas are given as follows: Tangential Component of Acceleration Normal Component of Acceleration a T = r (t) r (t) r (t) a N = r (t) r (t) r (t) We can calculate the curvature of a curve using the following formulas: 1

2 Curvature for y = f(x) Curvature for r(t) k(x) = f (x) [1+(f (x)) 2 ] 3 2 k(t) = r (t) r (t) r (t) 3 Projectile Motion Any problem with projectile motion can be solved once you find the position function r(t). This should almost always be your initial goal. Special Case: two dimensions, only gravity If the problem is in two dimensions (where the ground is the x-axis and the y-axis points up into the sky) with only gravity acting as acceleration, the position function always looks like: r(t) = x(t), y(t) = x 0 + v 0 cos(θ)t, y 0 + v 0 sin(θ)t 12 gt2 where (x 0, y 0 ) is the initial position, v 0 is the initial speed, and θ is the launching angle. In this case, the velocity function is given by: v(t) = r (t) = x (t), y (t) = v 0 cos(θ), v sin(θ) gt General Case In general, we could be in three dimensions (the xy-plane is the ground, the z-axis points into the sky) and there might be other components of acceleration alongside gravity. Usually, you know the constant acceleration function a(t) = a 1, a 2, a 3, and you know the initial velocity v(0) = v 1, v 2, v 3 and the initial position r(0) = r 1, r 2, r 3. To get the position function, start by integrating the acceleration vector with respect to t to get the velocity function: v(t) = a 1 t + C 1, a 2 t + C 2, a 3 t + C 3 To find the constants of integration C 1, C 2, C 3, plug in t = 0 and set this equal to the initial velocity: v(0) = C 1, C 2, C 3 = v 1, v 2, v 3 Therefore, v(t) = a 1 t + v 1, a 2 t + v 2, a 3 t + v 3 Next, integrate the velocity function to get position. Again, we find that the constants of integration are r 1, r 2, and r 3, using the initial position vector: 1 r(t) = 2 a 1t 2 + v 1 t + r 1, 1 2 a 2t 2 + v 2 t + r 2, 1 2 a 3t 2 + v 3 t + r 3 2

3 Then you can calculate whatever you need to. Note: the acceleration vector may not be constant. If this is the case, the position function will not look like the one above. The process for finding the position function is the same. Multivariable Functions Everything from here on out in 230 is just a generalization of single variable (differential) calculus. The main objects of study in single variable calculus are functions that look like f(x). They take in one number, and spit out another. Now, we consider functions that take in more than one number and spit another number. These look like f(x, y) or f(x, y, z). Domains of Functions Recall that the domain of a function f is all the numbers we can stick into f without running into trouble. If the function looks like f(x, y), the domain will be a subset of R 2, and if the function looks like f(x, y, z), the domain will be a subset of R 3. Here are some common questions you want to ask yourself when finding the domain of a function: - Do I ever divide by zero? - Do I ever take the square root of a negative number? - Do I ever take the natural log of a non-positive number? Etc. Level Curves, Level Surfaces, Contour plots Functions of multiple variables are hard to draw, so we resort to other tactics to help us visualize and think about them. The graph of a function z = f(x, y) is a surface:

4 Suppose that we slice through the surface with a horizontal plane at height k (this amounts to setting z = k, so f(x, y) = k); the intersection of the surface with the plane traces out a curve in the plane. The resulting curve defined by the equation f(x, y) = k is a level curve. If we find a bunch of level curves for different values of k and plot them all at once, we get a contour plot. For example, here is a contour plot of the function above: Each of those circles are level curves that represent slicing through the surface at a height of, 0.2, 0.8, etc. In other words, as you move along the circles in the domain, the function doesn t change. Given a function f(x, y, z) of three variables, we can do the same thing. The graph of f is in four dimensions, so we slice through with a three dimensional hyperplane by fixing a value of f i.e., setting f(x, y, z) = k. This traces out a surface in three dimensions, which we call a level surface. Multivariable Limits and Continuity In single variable calculus, we consider limits of functions as x approaches some point a in the domain. For the limit to exist, the one-sided limits have to be equal. In other words, we have to get the same limit no matter how we approached a. Given a function f(x, y) of two variables, the general ideas are the same. We can pick a point (a, b) in the domain and consider the limit of f as (x, y) approaches (a, b): lim f(x, y) For this limit to exist, we have to get the same value for every possible path of approach in the domain. If two different approaches two different values, then the limit does not exist. We can similarly consider functions of three variables: The same principles hold. lim f(x, y, z) (x,y,z) (a,b,c) 4

5 Limit Techniques Here are some common ways of evaluating the following limit: lim f(x, y) Examples of all of these techniques are in the example sheet. - Continuity / Plugging In If f is continuous at the point (a, b), then you can just plug the point in and calculate f(a, b). nothing bad happens (like dividing by 0), then that s the answer! If - Different Path Approaches If you can find two different approaches that give different results, then the limit does not exist. If the target point (a, b) is the origin (0, 0), some common paths are x = 0, y = 0, y = x, etc. For these paths, the limits become, respectively: lim f(0, y) lim y 0 f(x, 0) lim x 0 f(x, x) x 0 Note: you can use any path of approach; these are just some easy ones to check. - Factoring If f(x, y) is a fraction and yields an indeterminate form like 0 0, sometimes you can factor the top or bottom to conveniently cancel terms. Note: you can t use L Hopitals Rule with a multivariable limit: if you convert the limit to one variable, then you can use it, but not in general. - Conjugate If f(x, y) contains a term that looks like blah1 blah2, sometimes it helps to multiply the top and bottom by the conjugate, blah1 + blah2. - Squeeze Theorem / Comparison If you can find upper and lower bounds for f(x, y), evaluating the limits of those bounds can squeeze in the limit of f(x, y). This is sometimes useful if you have a trig function inside f(x, y). - Polar Coordinates If f(x, y) contains terms that look like x 2 + y 2 or x 2 + y 2 or something that reminds you of circles, and (a, b) = (0, 0), sometimes it is useful to convert f(x, y) into polar coordinates. Then the limit becomes: lim f(x, y) = lim f(r cos θ, r sin θ) (x,y) (0,0) r 0 5

6 Continuity of Multivariable Functions In single variable calculus, a function is continuous at a point a when lim x a f(x) = f(a), i.e., when the limit equals the value of the function. Similarly, a function f(x, y) is continuous at (a, b) if lim f(x, y) = f(a, b) Typically, you will be given a piecewise function defined as follows: g(x, y) (x, y) (a, b) f(x, y) = c (x, y) = (a, b) for some function g(x, y). This means that everywhere outside of the point (a, b), then function equals g(x, y). At the point (a, b), it equals c. If this function is to be continuous at the point (a, b), then we need: lim f(x, y) = f(a, b) lim g(x, y) = c So if lim g(x, y) = c, then f is continuous at (a, b), and if lim g(x, y) c, then f is discontinuous at (a, b). Partial Derivatives In single variable calculus, the derivative of a function f(x) represents how the function changes when we move in the positive x-direction by a little bit. In multivariable calculus, we have a choice as to which variable we can change. If f(x, y) is a function, the partial derivative of f with respect to x, denoted f x (x, y) or f x(x, y) represents how the function changes when we move in the positive x-direction. The partial derivative of f with respect to y is defined and denoted similarly. Calculating partial derivatives amounts to pretending that the other variable is a constant. So calculating the partial derivative of f with respect to f can be expressed as: d f(x, y) = f(x, C) x dx where C is a constant We can compute second partial derivatives by just taking the derivative again. Note: taking the partial with respect to x then y would be denoted: 2 f y x If the second partials of f exist and are continuous, then 2 f x y = 2 f y x or or f xy f yx = f xy 6

7 Interpreting Partial Derivatives with Contour Plots We can analyze the positivity and negativity of partial derivatives in the context of a contour plot. Some things to note: - Always move in the positive direction when looking at how the function changes. - When level curves get closer together, that means the change is becoming more extreme, either positively or negatively. So if the level curves are increasing in value and getting closer together, the second partial derivative in that direction will be positive. - Level curves represent paths of no change. If the direction you are moving in is tangent to a level curve, then the partial derivative with respect to that direction is 0. Tangent Planes and Linear Approximations In single variable calculus, we approximated functions by calculating tangent lines. In multivariable calculus, we do the same thing, but with tangent planes. Given a function f(x, y) of two variables, the formula for the tangent plane at the point (a, b) is: z = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) This is also called the linear approximation to f at (a, b). We can do the same thing with a function f(x, y, z) of three variables. approximation to f at (a, b, c) is: The formula for the linear L(x, y, z) = f(a, b, c) + f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) Note: these formulas look complicated, but they are natural generalizations of the tangent line / linear approximation formula in a single variable: y = f(a) + f (a)(x a) Tangents planes and linear approximations give us ways to approximate how functions change. Given a function z = f(x, y), we can start at a point (a, b) in the domain, change a little bit to (a + h 1, b + h 2 ), and calculate the difference in the function at these points. This is z: z = f(a + h 1, b + h 2 ) f(a, b) We can approximate z by using the tangent plane / linear approximation to f at the point (a, b). Suppose the linear approximation is given by T (x, y). Then dz is: dz = T (a + h 1, b + h 2 ) T (a, b) = T (a + h 1, b + h 2 ) f(a, b) 7

8 Quadratic Approximations An improvement can be made on linear approximations of two-variable functions using the second partial derivatives. Recall that the linear approximation to f(x, y) at (a, b) is: T (x, y) = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) The quadratic approximation to f at (a, b) is given by adding some new terms to the linear approximation: Q(x, y) = T (x, y) [ fxx (a, b)(x a) 2 + 2f xy (a, b)(x a)(y b) + f yy (a, b)(y b) 2] This is also called the second-order Taylor polynomial to f at (a, b). The Chain Rule The single variable chain rule says that if f(x) is a function that depends on x, and x(t) is a function that depends on t, then if we want to know how f changes when t changes, we calculate df dt = df dx dx dt. This is represented by the diagram on the right, where the total change is calculated by multiplying each individual level of change. The chain rule in multiple variables extends this diagram to have (possibly) multiple legs. To find the total change of f with respect to a variable, you add each leg of the change that involves that variable. For example, in the diagram below, f x t f x y u v u v f(x, y) depends on x and y, and x and y both depend on u and v. If we want to find f v, there are two legs of change to find. These are highlighted in red: f x y u v u v Hence, f v = f x x v + f y y v 8

9 Gradients and Directional Derivatives The gradient of a function f(x, y) (respectively f(x, y, z)) is: f f = x, f f f = y x, f y, f z The gradient represents the direction of fastest increasing rate of change. The actual rate of change in that direction is f. The direction of fastest decreasing rate of change is f. The gradient f is perpendicular / orthogonal / normal to level curves f(x, y) = k and level surfaces f(x, y, z) = k. Therefore, we can calculate equations of tangent planes to level surfaces by calculating the gradient and using that as a normal vector. Given a unit vector u, the directional derivative of f in the direction u is: D u f = u f This represents the rate of change in the u direction. Note: D i f = f x D j f = f y 9