Differentiability and Tangent Planes October 2013
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1 Differentiability and Tangent Planes October 2013
2 Differentiability in one variable. Recall for a function of one variable, f is differentiable at a f (a + h) f (a) lim exists and = f (a) h 0 h f (a + h) f (a) hf (a) lim = 0 h 0 ( h f (x) f (a) + f (a)(x a) ) lim = 0 x a x a The local linearization of f at x = a is L(x) = f (a) + f (a)(x a)
3 Differentiability in one variable, continued. If lim x a f (x) L(x) x a = 0, then: L(x) is a good approximation to f (x): f (x) L(x) for x near a Not only f (x) L(x) is small, but it s actually small compared to x a, which is already small! Also, y = L(x) is an equation for the tangent line to the graph of f at x = a
4 Local linearization. For a function f (x, y) of two (or more) variables, if f x (a, b) and f y (a, b) exist, the local linearization of f at (a, b) is L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b). (a, b)
5 Local linearization. For a function f (x, y) of two (or more) variables, if f x (a, b) and f y (a, b) exist, the local linearization of f at (a, b) is L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b). f x (a, b) x (a, b) x (a + x, b)
6 Local linearization. For a function f (x, y) of two (or more) variables, if f x (a, b) and f y (a, b) exist, the local linearization of f at (a, b) is L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b). f y (a, b) y f x (a, b) x (a, b + y) y x (a, b) (a + x, b)
7 Local linearization. For a function f (x, y) of two (or more) variables, if f x (a, b) and f y (a, b) exist, the local linearization of f at (a, b) is L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b). f y (a, b) y f x (a, b) x + f y (a, b) y f x (a, b) x (a, b + y) y (a + x, b + y) x (a, b) (a + x, b)
8 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): f (1, 2) = = 3, f x (x, y) =, f x (1, 2) =, f y (x, y) =, f y (1, 2) = so L(x, y) = + (x ) + (y )
9 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): f (1, 2) = = 3, f x (x, y) = 1 2 (x + y 3 ) 1/2 (1), f x (1, 2) =, f y (x, y) =, f y (1, 2) = so L(x, y) = + (x ) + (y )
10 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): f (1, 2) = = 3, f x (x, y) = 1 2 (x + y 3 ) 1/2 (1), f x (1, 2) = = 1 6, f y (x, y) =, f y (1, 2) = so L(x, y) = + (x ) + (y )
11 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): f (1, 2) = = 3, f x (x, y) = 1 2 (x + y 3 ) 1/2 (1), f x (1, 2) = = 1 6, f y (x, y) = 1 2 (x + y 3 ) 1/2 (3y 2 ), f y (1, 2) = so L(x, y) = + (x ) + (y )
12 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): so f (1, 2) = = 3, f x (x, y) = 1 2 (x + y 3 ) 1/2 (1), f x (1, 2) = = 1 6, f y (x, y) = 1 2 (x + y 3 ) 1/2 (3y 2 ), f y (1, 2) = = 2 L(x, y) = + (x ) + (y )
13 Example: Local linearization. Example: f (x, y) = x + y 3, local linearization at (1, 2): so f (1, 2) = = 3, f x (x, y) = 1 2 (x + y 3 ) 1/2 (1), f x (1, 2) = = 1 6, f y (x, y) = 1 2 (x + y 3 ) 1/2 (3y 2 ), f y (1, 2) = = 2 L(x, y) = (x 1) + 2(y 2). 6
14 Differentiability. Definition f is locally linear at (a, b) if f x (a, b) and f y (a, b) exist, and lim (x,y) (a,b) f (x, y) L(x, y) = 0. (1) (x a)2 + (y b) 2 f is differentiable at (a, b) if there exists a linear function L(x, y) such that (1) holds e.g., if f is locally linear at (a, b). Theorem If f x and f y exist and are continuous on an open disk D, then f is differentiable on D. (It s possible for f to be differentiable even if f x and f y don t exist, or are not continuous! See textbook, Exercises and )
15 Tangent Planes. If f is differentiable at (a, b) then z = L(x, y) is an equation for the tangent plane to the graph of f at the point (a, b). Example: z = (x 1) + 2(y 2) is the tangent plane 6 to the graph of z = x + y 3 at the point (1, 2). Normal vector:
16 Tangent Planes. If f is differentiable at (a, b) then z = L(x, y) is an equation for the tangent plane to the graph of f at the point (a, b). Example: z = (x 1) + 2(y 2) is the tangent plane 6 to the graph of z = x + y 3 at the point (1, 2). 1 Normal vector:, 2, 1 6
17 Tangent Planes. If f is differentiable at (a, b) then z = L(x, y) is an equation for the tangent plane to the graph of f at the point (a, b). Example: z = (x 1) + 2(y 2) is the tangent plane 6 to the graph of z = x + y 3 at the point (1, 2). 1 Normal vector:, 2, 1 6 Normal vector = f x, f y, 1.
18 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces
19 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces The tangent lines to the traces are z = f x (a, b)(x a), y = b z = f y (a, b)(y b), x = a
20 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces The tangent lines to the traces are z = f x (a, b)(x a), y = b x, y, z = t, b, f x (a, b)(t a) z = f y (a, b)(y b), x = a
21 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces The tangent lines to the traces are z = f x (a, b)(x a), y = b x, y, z = t, b, f x (a, b)(t a) z = f y (a, b)(y b), x = a x, y, z = a, t, f y (a, b)(t b)
22 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces The tangent lines to the traces are z = f x (a, b)(x a), y = b x, y, z = t, b, f x (a, b)(t a) z = f y (a, b)(y b), x = a x, y, z = a, t, f y (a, b)(t b) They have direction vectors 1, 0, f x (a, b), 0, 1, f y (a, b)
23 Normal vector: Geometric explanation. Why is f x, f y, 1 the normal vector? Tangent plane has to contain tangent lines of traces The tangent lines to the traces are z = f x (a, b)(x a), y = b x, y, z = t, b, f x (a, b)(t a) z = f y (a, b)(y b), x = a x, y, z = a, t, f y (a, b)(t b) They have direction vectors 1, 0, f x (a, b), 0, 1, f y (a, b) The tangent plane contains those two direction vectors, so a normal vector is n = 0, 1, f y (a, b) 1, 0, f x (a, b) = = f x (a, b), f y (a, b), 1.
24 Linear Approximation. If f is differentiable at (a, b) then for (x, y) near (a, b), f (x, y) L(x, y).
25 Worksheet. Worksheet #1 3
26 Clicker Question: Let f (2, 3) = 7, f x (2, 3) = 1, and f y (2, 3) = 4. Then the tangent plane to the surface z = f (x, y) at the point (2, 3) is A. z = 7 x + 4y B. x 4y + z + 3 = 0 C. x + 4y + z = 7 D. x + 4y + z + 3 = 0 E. z = 17 + x 4y receiver channel: 41 session ID: bsumath275
27 Clicker Question: The figure below shows the level curves of f (x, y). The tangent plane approximation to f (x, y) at the point P = (x 0, y 0 ) is f (x, y) c + m(x x 0 ) + n(y y 0 ). What are the signs of c, m, and n? y A. c > 0, m > 0, n > B. c < 0, m > 0, n < 0 C. c > 0, m < 0, n > 0 P x D. c < 0, m < 0, n < 0 E. c > 0, m > 0, n < 0 receiver channel: 41 session ID: bsumath275
28 Clicker Question: Suppose f x (3, 4) = 5, f y (3, 4) = 2, and f (3, 4) = 6. Assuming the function is differentiable, what is the best estimate for f (3.1, 3.9) using this information? A. 6.3 B. 9 C. 6 D. 6.7 receiver channel: 41 session ID: bsumath275
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