The Differential df, the Gradient f, & the Directional Derivative Dû f sec 14.4 (cont), Goals. Warm-up: Differentiability. Notes. Notes.

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1 The Differential df, the Gradient f, & the Directional Derivative Dû f sec 14.4 (cont), March 2014 Goals. We will: Define the differential df and use it to approximate changes in a function s value. Define and practice computing the gradient of a function, and state some important properties of the gradient. Define and practice computing directional derivatives. (Didn t make it we ll start here on Wednesday 3/12.) Warm-up: Differentiability. Which of the following statements best reflects the way you think about differentiability of a function f (x, y)? 1. There is a tangent plane. 2. There is a linear function that approximates the function. 3. There isn t a bad point cone point, break, singularity, etc. on the graph of the function. 4. I don t understand the question. receiver channel: 41 session ID: mth275

2 Warm-up: Linearization. Compute the linearization of the function V (r, h) = πr 2 h. Click in when finished. (We will use this later in the class.) 1. Finished. receiver channel: 41 session ID: mth275 The Differential df If f (x, y) is differentiable at the point (a, b), then the change in the function s value at points close to (a, b) can be approximated using the local linearization: f = f (x, y) f (a, b) L(x, y) f (a, b) = f (y b) This approximation is called the differential, denoted df. It is common practice to replace (x a) with x or dx, and (y b) with y or dy in the differential: dy Summary If the partial derivatives of f are continuous on an open neighborhood of (a, b): Local linearization approximates the function value: L(x, y) = f (y b) + f (a, b) Tangent plane the graph of L: z = L(x, y) Differential approximates changes in the function value: dy

3 For Functions of Three Variables Three variables f (x, y, z) at (a, b, c): Local linearization: L(x, y, z) = f (y b) x (a,b,c) y (a,b,c) + f (z c) + f (a, b, c). z (a,b,c) Tangent space (the graph of L looks like R 3 ): w = L(x, y, z) Differential: dy + f dz x (a,b,c) y (a,b,c) z (a,b,c) Linear Approximation vs. the Differential Again: The local linearization L approximates the function value: L f. The tangent space (plane, for a function of two variables) is the graph of L. The differential df approximates the change in the function s value: df f. Example: Linear Approximation vs. the Differential Suppose T (φ, θ, z), a differentiable function, represents the temperature in Idaho at a fixed time, as a function of latitude φ, longitude θ, and elevation above sea-level z. A temperature measurement is taken in Boise. All thermometers in Nampa are broken. To answer the question, What s an estimate of the temperature in Nampa?, use the local linearization L(φ, θ, z), centered at Boise. To answer the question, How many degrees warmer or colder is it in Nampa than in Boise?, use the differential dt, centered at Boise.

4 Example: Volume of a Cylinder The volume of a right cylinder of height h and base a circle of radius r is given by the equation V = πr 2 h. For the following, assume a cylinder with radius r 0 = 2 cm and height h 0 = 3 cm. Use the local linearization to estimate the volume when the radius is 2.01 cm and the height is 2.99 cm. Use the differential to estimate the change in volume if the radius is increases by 0.01 cm, while simultaneously, the height decreases by 0.01 cm. If the radius is increased by 0.01 cm, by how much must the height change in order for the volume to remain constant? The Gradient Vector Suppose the first partial derivatives of f exist. The gradient of f (denoted f ) is the vector field whose i th component is the i th partial derivative of f. In particular: If f is a function of two variables x and y: f = f x î + f y ĵ If f is a function of three variables x, y and z: f = f x î + f y ĵ + f z ˆk Example: the Gradient Compute the gradient of the function f (x, y) = x 2 + y 2 in general, then at the points P = (1, 0) and Q = (1, 1). What do you notice about the relationship between the gradient and the level curves of f? What do you notice about the direction of f relative to the graph of f?

5 Properties of the Gradient Suppose the first partial derivatives of f (x, y) are continuous. Then f is differentiable and: The gradient f is orthogonal to the level curves of f. The slope of the tangent plane is greatest in the direction of f (the gradient points in the direction in which the function is increasing most rapidly). The value of the slope of the tangent plane in the direction of f is f (the magnitude of the gradient gives the value of the greatest rate of increase of the function at a point). (These will be proved later using directional derivatives.)