Math%126%Midterm%2%Study%Guide%Grace%Qiu%%1. Vector functions ! Computing curvature " k(t) = |!!! !!!!!!! |!!!! " Value of k(t) gives the numerical measure of how “tight” a curve is at time “t”. " High curvature = tighter curve " Lower curvature = looser curve " Curvature = !!"#!"# Radius = !!"#$%&"#' " If k(t) is not in terms of a variable, that means the curve is constant. ! Finding unit tangent vector (unit vector perpendicular to r(t) at time “t”) " T(t) = !!! !|!!! | " The unit tangent vector has a magnitude of 1. " For every value of “t”, T’(t) is perpendicular to T(t). ! Finding normal vector (unit vector perpendicular to T(t) at time “t”) " N(t) = !!! !|!!! | " If you think of the curve defined by r(t) as having a circle of best fit at “t”… ! Finding bi-normal vector (unit vector bi-normal to r(t)) " B(t) = T(t) x N(t) " Think of T(t), N(t), and B(t) as a frame, like a car on a rollercoaster. This frame moves along the r(t) curve (the roller coaster) as “t” varies. In this analogy, T(t) is the direction the car is heading, N(t) is pointing towards the inside of the curve you are travelling on, and B(t) is pointing up/down from your position. (*Think of the diagram above as a bird’s eye view of a rollercoaster.) T(t)%N(t)%t%r(t)%Math%126%Midterm%2%Study%Guide%Grace%Qiu%%! Finding normal plane " This is the plane that contains B(t) and N(t). T(t) is perpendicular to this plane. " In the rollercoaster analogy, this plane is the front side of the car. " Process (given r(t) and a value of “t”): 1. Find r’(t). 2. Using this, find T(t). 3. Find a point on r(t), so plug in the value of “t” into the original r(t) equation. 4. The normal plane contains this point and T(t) is perpendicular to this plane, so combine the two into a plane equation. ! Finding osculating plane " This is the plane that contains T(t) and N(t). B(t) is perpendicular to this plane. " In the rollercoaster analogy, this plane is the floor of the car. " Process (given r(t) and a value of “t”): 1. Find r’(t). 2. Using this, find T(t). 3. Using this, find N(t). ***Here on out, it is safe to plug in “t”*** 4. Using T(t) and N(t), find B(t). 5. Find a point on r(t), so plug in the value of “t” into the original r(t) equation. 6. The osculating plane contains this point and B(t) is perpendicular to this plane, so combine the two into a plane equation. ! Position/Velocity/Speed/Acceleration of an object on the curve of r(t) " r’(t) = velocity " | r’(t) | = speed (magnitude with no direction.) " r’’(t) = acceleration (acc. lives in the osculating plane!) " aN = |!!! !!!!!!! ||!!! | " aT = !!!! !!!!!(!)|!! ! | Recall: the equation of a plane consists of a point on the plane and a vector that is perpendicular to the plane. You can plug in the value for “t” to make finding B(t) easier. However, DO NOT plug in until you have both T(t) and N(t). Recall: the equation of a plane consists of a point on the plane and a vector that is perpendicular to the plane. In a straight line, acc. is all in the tangential direction. In a circle, acc. is all in the normal direction. T(t)%N(t)%a(t)=aNN(t)%+%aTT(t)%a(t) is made up of a normal component (aN) and a tangential component (aT). To find a(t), you need both components.Math%126%Midterm%2%Study%Guide%Grace%Qiu%%2. Two-variable functions (z = f(x,y)) ! Draw and interpret level curves and contour maps " Contour map: set of level curves. " To get a single level curve, find constant z of a certain (x,y). " This is a bird’s eye view of a function f(x,y) looking down at the xy plane. The “k” values are values of z, or the height at that x and y.% ! Compute and interpret partial derivatives " Suppose z = f(x,y). # fx (x,y) : to find the partial derivative with respect to x, pretend y is just another constant and derive normally (keeping y in the equation as a constant.) # fy (x,y) : To find the partial derivative with respect to y, pretend x is just another constant and derive normally (keeping x in the equation as a constant.) # A two-variable function has two partial derivatives. As such, it has four partial second derivatives as each partial derivative has two derivatives (one with respect to x, one with respect to y.) These are denoted as fxx (x, y) , fxy (x, y) , fyx (x,y) , and fyy (x,y). ! Implicit differentiation " To find dy/dx: # Consider y a function in terms of x. In other words, y = y(x). # Derive the entire equation with respect to x. This should result in a dy/dx appearing in the equation, as deriving y(x) gives dy/dx. # Ex: Given 25x2 + y2 = 109, find dy/dx. If you walk along the surface in the direction of increasing z, then you’re walking uphill. If you walk along a single level curve, your altitude isn’t changing. Think of a topography graph. When level curves are close together, the surface is steep. If they are further apart, they are less steep.Math%126%Midterm%2%Study%Guide%Grace%Qiu%% " To find dx/dy: # Consider x a function in terms of y. In other words, x = x(y). # Derive the entire equation with respect to y. This should result in a dx/dy appearing in the equation, as deriving x(y) gives dx/dy.% ! Find the equation of a tangent plane " z = fx (xo, yo)(x - xo) + fy (xo, yo)(y - yo) + f(xo, yo) " Normal vector N(t) of this tangent plane = < fx (xo, yo), fy (xo, yo), -1> " Process: 1. Find f(xo, yo). This gives you the z coordinate for the given x and y, giving you a point on the function with coordinates (x, y, z) 2. Find fx (xo, yo) and plug in given x and y values. 3. Find fy (xo, yo) and plug in given x and y values. 4. Plug these values into the equation of z above to get the equation of the plane. ! Use linear approximation (linearization) to approximate a specific value of z " T(x, y) = fx (xo, yo)(x - xo) + fy (xo, yo)(y - yo) + f(xo, yo) " This is the same equation for the tangent plane. " To find a value at x and y (where x and y are values close to the values of xo and yo) plug in original points xo and yo and new points x and y. ! Compute and interpret the total differential " dz = !"!"!𝑑𝑥 + !!"!"!!𝑑𝑦 " Gives an approximation of the change in z if x changes by dx and y changes by dy. ! Find all critical points " Critical numbers in single variable functions are the values of x that make the derivative 0 (a horizontal tangent