Topic 2.3: Tangent Planes, Differentiability, and Linear Approximations Textbook: Section 14.4
Warm-Up: Graph the Cone & the Paraboloid paraboloid f (x, y) = x 2 + y 2 cone g(x, y) = x 2 + y 2 Do you notice anything significantly different about these two graphs?
Warm-Up: Partial Derivatives Which one of the following is incorrect notation for expressing the partial derivative of a function f (x, y) with respect to x? A. f x (x, y) B. C. f x df dx D. All three are correct notation. E. If I answered, I would be guessing.
Warm-Up: Partial Derivatives f (x, y) = e x tan 2 x + yx 2 Half of the room will compute f x (x, y) (differentiate with respect to x). The other half will compute f y (x, y) (differentiate with respect to y). Click in when you are finished. A. I have finished computing f xy (x, y) (differentiate with respect to x first). B. I have finished computing f yx (x, y) (differentiate with respect to y first).
Warm-Up: Clairaut s Theorem The easiest way to compute the mixed second partial derivative of the function: is: f (x, y) = e x tan 2 x + yx 2 A. Differentiate with respect to x first, then y. B. Differentiate with respect to y first, then x. C. It doesn t matter. D. I don t know how to answer this question.
Big Ideas In Calc I, differentiability of a function y = f (x) (single variable) was associated with the existence of a well-defined tangent line. In Calc III, differentiability of a function f (x, y) (two variables) is associated with the existence of a well-defined tangent plane. The tangent plane of f at a point (a, b, f (a, b)) is the graph of a linear function L, which can be used to approximate values for f at points near (a, b). If f (x, y, z) is a function of three variables, the same ideas apply, but the tangent space is three-dimensional.
Big Question for Today: What does it mean for a function f (x, y) to be differentiable?
Recall fr Calc I: Differentiability of a Function f (x) If f (x) is differentiable at a point a, then: The graph y = f (x) has a well-defined tangent line at the point ( a, f (a) ). The tangent line is the best linear approximation of f (x) at a. As you zoom in on the point ( a, f (a) ), the tangent line looks more and more like the graph of f (x). The slope of the tangent line is the derivative f (a). There s a big difference between y = f (x) and z = f (x, y). For z = f (x, y), there are infinitely many tangent lines, and each (usually) has a different slope!
Differentiability of a Function f (x, y) If f (x, y) is differentiable at a point (a, b), then: The graph z = f (x, y) has a well-defined tangent plane at the point ( a, b, f (a, b) ). The tangent plane is the best linear approximation of f (x, y) at (a, b). As you zoom in on the point ( a, b, f (a, b) ), the tangent plane looks more and more like the graph of f (x, y). (For function of three variables, differentiability means there is a well-defined 3-d tangent space, and so on in higher dimensions.)
Tangent Planes, Tangent Lines, and Derivatives If f (x, y) is differentiable at a point (a, b), then it has a well-defined tangent plane at the point (a, b). The structure of this tangent plane is such that: The tangent plane to f (x, y) at ( a, b ) is made up of infinitely many tangent lines passing through the point ( a, b, f (a, b) ). The slopes of these tangent lines are the directional derivatives; the slopes of the tangent lines in the x and y directions are the partial derivatives f x (a, b) and f y (a, b). The slopes of the tangent lines in the x and y directions are the partial derivatives f x (a, y) and f y (a, b).
Example: the Cone vs the Paraboloid paraboloid f (x, y) = x 2 + y 2 cone g(x, y) = x 2 + y 2 Do you notice anything significantly different about these two graphs?
Partial Derivatives and Tangent Planes How can you tell whether a function is differentiable and has a tangent plane? Theorem: If the first partial derivatives f x (x, y) and f y (x, y) exist and are continuous in a neighborhood of a point (a, b), then: f is differentiable at (a, b), and: f has a well-defined tangent plane at ( a, b ). The analogous result is true for functions of more than two variables. If all first partial derivatives exist and are continuous in a neighborhood of a point P, then the function is differentiable and has a well-defined tangent space.
Partial Derivatives and Tangent Planes: Examples Compute the partial derivatives and evaluate them at (a, b) = (0, 0) for: The paraboloid f (x, y) = x 2 + y 2. The cone g(x, y) = x 2 + y 2. How can you tell from the partial derivatives that the cone is not differentiable at the origin?
Equation of Tangent Spaces For a function of two variables: If f x and f y are continuous on an open neighborhood of (a, b), the equation of the tangent plane to f at the point ( a, b, f (a, b) ) is: z = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b). For a function of three variables: If f x, f y, and f z are continuous on an open neighborhood of the point (a, b, c), the equation of the tangent plane to f at the point ( a, b, c, f (a, b, c) ) is: z = f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) + f (a, b, c).
Example: Tangent Plane Find the equation of the tangent planes for: Algorithm: f (x, y) = xy + x + 1 at the point ( 1, 2, f (1, 2) ) Compute partial derivatives. Evaluate the partial derivatives f x and f y and the function f at the point (1, 2). Use your answer from A and B in the equation for the tangent plane: z = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b).
Clicker Question: Tangent Plane What is the coefficient of the x-variable in your equation of the tangent plane to f (x, y) = xy + x + 1 at the point (1, 2, 4)? Enter the value for this coefficient. Enter 500 if you did not finish the computation.
Clicker Question: Tangent Plane What is the constant term in your equation of the tangent plane to f (x, y) = xy + x + 1 at the point (1, 2, 4)? Enter the value of this constant term. Enter 500 if you did not finish the computation.
The Local Linearization L The local linearization L of a differentiable function f at a point P is the linear function whose graph is the tangent plane. The local linearization L can be used to approximate function values near the point of tangency. It answers the question: What is the approximate value of f for points near P? In Calc I, the local linearization in the first degree Taylor polynomial T 1 (x) aka the tangent line approximation.
Equation of the Local Linearization for Functions of Two or Three Variables If f (x, y) is a differentiable function of two variables: L(x, y) = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b). If f (x, y, z) is a differentiable function of three variables: L(x, y, z) = f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) + f (a, b, c). Since the tangent plane is the graph of the linearization L, they are both defined by the same equation.
Example: Using the Linearization to Approximate Function Values Use linear approximation to estimate the value (4.01) 3 (1.02) 2. (First thing to do: figure out what function is being approximated, and where to center the approximation!) Find the percent error in the approximation (rounded to 2 decimal places).