MATH 19520/51 Class 6

Transcription

1 MATH 19520/51 Class 6 Minh-Tam Trinh University of Chicago

2 1 Review partial derivatives. 2 Review equations of planes. 3 Review tangent lines in single-variable calculus. 4 Tangent planes to graphs of functions of two variables.

3 Review of Partial Derivatives Find the 1 st -order partial derivatives of (1) 1 f (x, y, z) = x + y + 1. z

4 Review of Partial Derivatives Find the 1 st -order partial derivatives of (1) 1 f (x, y, z) = x + y + 1. z You should get: f x = 1 (2) f y = (y + 1 z ) 2 f z = (y + 1 z ) 2 z 2 = (yz + 1) 2

5 Review of Lines in the (x, y)-plane In precalculus, you learned one way to write equations for lines in the (x, y)-plane: (3) y = mx + b is a line with slope m and y-intercept b.

6 Review of Lines in the (x, y)-plane In precalculus, you learned one way to write equations for lines in the (x, y)-plane: (3) y = mx + b is a line with slope m and y-intercept b. (The y-intercept indicates that the line passes through (x, y) = (0, b).)

7 Review of Lines in the (x, y)-plane In precalculus, you learned one way to write equations for lines in the (x, y)-plane: (3) y = mx + b is a line with slope m and y-intercept b. (The y-intercept indicates that the line passes through (x, y) = (0, b).) What about a line that has slope m in the positive x-direction and passes through the point (x, y) = (a, b)?

8 The line of slope m that passes through (a, b) is given by (4) y b = m(x a). Check that (x, y) = (a, b) does in fact solve this equation!

9 The line of slope m that passes through (a, b) is given by (4) y b = m(x a). Check that (x, y) = (a, b) does in fact solve this equation! Example What is the line through (0, 0) of slope 1? What is the line through (3, 4) of slope 0? What is the line through (0, b) of slope m?

10 The line of slope m that passes through (a, b) is given by (4) y b = m(x a). Check that (x, y) = (a, b) does in fact solve this equation! Example What is the line through (0, 0) of slope 1? y = x What is the line through (3, 4) of slope 0? What is the line through (0, b) of slope m?

11 The line of slope m that passes through (a, b) is given by (4) y b = m(x a). Check that (x, y) = (a, b) does in fact solve this equation! Example What is the line through (0, 0) of slope 1? y = x What is the line through (3, 4) of slope 0? y = 4 What is the line through (0, b) of slope m?

12 The line of slope m that passes through (a, b) is given by (4) y b = m(x a). Check that (x, y) = (a, b) does in fact solve this equation! Example What is the line through (0, 0) of slope 1? y = x What is the line through (3, 4) of slope 0? y = 4 What is the line through (0, b) of slope m? y = mx + b

13 Review of Planes in (x, y, z)-space Warning! In the (x, y)-plane, y is the vertical coordinate. But in (x, y, z)-space, z is the vertical coordinate.

14 Review of Planes in (x, y, z)-space Warning! In the (x, y)-plane, y is the vertical coordinate. But in (x, y, z)-space, z is the vertical coordinate. We can find the equation of a line in the (x, y)-plane from knowing a point on it and its slope in the x-direction.

15 Review of Planes in (x, y, z)-space Warning! In the (x, y)-plane, y is the vertical coordinate. But in (x, y, z)-space, z is the vertical coordinate. We can find the equation of a line in the (x, y)-plane from knowing a point on it and its slope in the x-direction. Can we find the equation of a plane in (x, y, z)-space from knowing a point on it and its slopes in the x- and y-directions?

16 The plane of slope m x in the x-direction and slope m y in the y-direction that passes through (a, b, c) is given by (5) z c = m x (x a) + m y (y b).

17 The plane of slope m x in the x-direction and slope m y in the y-direction that passes through (a, b, c) is given by (5) z c = m x (x a) + m y (y b). Example What is the plane through (0, 0, 0) of slope 3 in the x-direction and flat in the y-direction? What is the plane through (3, 4, 12) that is completely flat in both the x- and y-directions? What is the plane that contains the lines u = (1, 1, 1) + t(1, 0, 3) and u = (1, 1, 1) + t(0, 1, 2)?

18 The plane of slope m x in the x-direction and slope m y in the y-direction that passes through (a, b, c) is given by (5) z c = m x (x a) + m y (y b). Example What is the plane through (0, 0, 0) of slope 3 in the x-direction and flat in the y-direction? z = 3x What is the plane through (3, 4, 12) that is completely flat in both the x- and y-directions? What is the plane that contains the lines u = (1, 1, 1) + t(1, 0, 3) and u = (1, 1, 1) + t(0, 1, 2)?

19 The plane of slope m x in the x-direction and slope m y in the y-direction that passes through (a, b, c) is given by (5) z c = m x (x a) + m y (y b). Example What is the plane through (0, 0, 0) of slope 3 in the x-direction and flat in the y-direction? z = 3x What is the plane through (3, 4, 12) that is completely flat in both the x- and y-directions? z = 12 What is the plane that contains the lines u = (1, 1, 1) + t(1, 0, 3) and u = (1, 1, 1) + t(0, 1, 2)?

20 The plane of slope m x in the x-direction and slope m y in the y-direction that passes through (a, b, c) is given by (5) z c = m x (x a) + m y (y b). Example What is the plane through (0, 0, 0) of slope 3 in the x-direction and flat in the y-direction? z = 3x What is the plane through (3, 4, 12) that is completely flat in both the x- and y-directions? z = 12 What is the plane that contains the lines u = (1, 1, 1) + t(1, 0, 3) and u = (1, 1, 1) + t(0, 1, 2)? z 1 = 3(x 1) 2(y 1)

21 Review of Tangent Lines The red line is tangent to the black curve at the intersection point on the left, which is called the point of tangency.

22 A.gif showing how the tangent line to a differentiable curve changes as the point of tangency changes: sliding_derivative_line.gif Note: If a function is not differentiable at some point, then its graph might not have a tangent line there. Draw this.

23 If f (x) is differentiable and (a, b) is a point on the graph y = f (x), then what s the equation of the tangent line at (a, b)?

24 If f (x) is differentiable and (a, b) is a point on the graph y = f (x), then what s the equation of the tangent line at (a, b)? The slope of f at x = a is its derivative f (a). So the tangent line has equation (6) y b = f (a)(x a).

25 Example Find the tangent line to y = 1 9 x2 at the point where x = 6.

26 Example Find the tangent line to y = 1 9 x2 at the point where x = 6. The desired point of tangency is (6, 4). We compute (7) dy dx = 2 9 x, so the slope at that point is = 4 3.

27 Example Find the tangent line to y = 1 9 x2 at the point where x = 6. The desired point of tangency is (6, 4). We compute (7) dy dx = 2 9 x, so the slope at that point is = 4 3. The equation of the line is (8) y 4 = 4 (x 6). 3

28 The black curve is y = 1 9 x2. The green line is y 4 = 4 3 (x 6).

29 Example Find the tangent line to y = x sin x at the point where x = 2π.

30 Example Find the tangent line to y = x sin x at the point where x = 2π. The point in question is (2π, 2π). We compute (9) dy dx = 1 2 cos x, x so the slope at that point is 1 2 2π 1.

31 Example Find the tangent line to y = x sin x at the point where x = 2π. The point in question is (2π, 2π). We compute (9) dy dx = 1 2 cos x, x 1 so the slope at that point is 2 1. The line is 2π y ( ) 1 (10) 2π = 2 2π 1 (x 2π).

32 The black curve is y = x sin x. The green line is y ( ) 1 2π = 2 1 (x 2π). 2π

33 Review of Graphs of 2-Variable Functions Functions of the form y = f (x) give us curves in the (x, y)-plane.

34 Review of Graphs of 2-Variable Functions Functions of the form y = f (x) give us curves in the (x, y)-plane. Functions of the form z = f (x, y) give us surfaces in (x, y, z)-space.

35 Review of Graphs of 2-Variable Functions Functions of the form y = f (x) give us curves in the (x, y)-plane. Functions of the form z = f (x, y) give us surfaces in (x, y, z)-space. Which is z = x 2 + y 2 and which is z = x 2 y 2?

36 Tangent Planes to Graphs of 2-Variable Functions Curves in the (x, y)-plane can have tangent lines.

37 Tangent Planes to Graphs of 2-Variable Functions Curves in the (x, y)-plane can have tangent lines. Surfaces in (x, y, z)-space can have tangent planes.

38 If g(x, y) is differentiable and (a, b, c) is a point on the graph z = g(x, y), what s the equation of the tangent plane at (a, b, c)?

39 If g(x, y) is differentiable and (a, b, c) is a point on the graph z = g(x, y), what s the equation of the tangent plane at (a, b, c)? At (x, y) = (a, b), the slope in the x-direction is g x (a, b) and the slope in the y-direction is g y (a, b), so we obtain (11) z c = g x (a, b)(x a) + g y (a, b)(y b).

40 If g(x, y) is differentiable and (a, b, c) is a point on the graph z = g(x, y), what s the equation of the tangent plane at (a, b, c)? At (x, y) = (a, b), the slope in the x-direction is g x (a, b) and the slope in the y-direction is g y (a, b), so we obtain (11) z c = g x (a, b)(x a) + g y (a, b)(y b). This implies (12) f (x, y) c g x (a, b)(x a) + g y (a, b)(y b) near the point (x, y) = (a, b). Stewart calls this a linear approximation.

41 Example Find the tangent plane to the surface formed by z = x 2 + y 2 at the point where (x, y) = (3, 0).

42 Example Find the tangent plane to the surface formed by z = x 2 + y 2 at the point where (x, y) = (3, 0). Since = 9, the point of tangency is (3, 0, 9). We have (13) (14) z x (3, 0) = (2x) (x,y)=(3,0) = 6, z y (3, 0) = (2y) (x,y)=(3,0) = 0.

43 Example Find the tangent plane to the surface formed by z = x 2 + y 2 at the point where (x, y) = (3, 0). Since = 9, the point of tangency is (3, 0, 9). We have (13) (14) z x (3, 0) = (2x) (x,y)=(3,0) = 6, z y (3, 0) = (2y) (x,y)=(3,0) = 0. Therefore the tangent plane is (15) z 9 = 6(x 3) + 0(y 0), which simplifies to z 9 = 6(x 3), or just z = 6x 9.

44 Differentials In Stewart s terminology, the (total) differential of z = f (x 1,..., x n ) at (x 1,..., x n ) = (a 1,..., a n ) is (16) dz = f x1 (a 1,..., a n ) dx f xn (a 1,..., a n ) dx n. Plugging in dx i = x i a i and dz = z f (a 1,..., a n ) shows that this formula generalizes the equation for a tangent line/plane.

45 Differentials In Stewart s terminology, the (total) differential of z = f (x 1,..., x n ) at (x 1,..., x n ) = (a 1,..., a n ) is (16) dz = f x1 (a 1,..., a n ) dx f xn (a 1,..., a n ) dx n. Plugging in dx i = x i a i and dz = z f (a 1,..., a n ) shows that this formula generalizes the equation for a tangent line/plane. Example Find the differential of z = x 2 y 2 at (x, y) = (2, 2).

46 Differentials In Stewart s terminology, the (total) differential of z = f (x 1,..., x n ) at (x 1,..., x n ) = (a 1,..., a n ) is (16) dz = f x1 (a 1,..., a n ) dx f xn (a 1,..., a n ) dx n. Plugging in dx i = x i a i and dz = z f (a 1,..., a n ) shows that this formula generalizes the equation for a tangent line/plane. Example Find the differential of z = x 2 y 2 at (x, y) = (2, 2). If f (x, y) = x 2 y 2, then f x (2, 2) = 4 and f y (2, 2) = 4. Thus the differential dz equals (17) 4 dx 4 dy.

47 If dx i is the error bound on a measurement of x i, then dz is the largest possible error in z resulting from measurements of x 1,..., x n.

48 If dx i is the error bound on a measurement of x i, then dz is the largest possible error in z resulting from measurements of x 1,..., x n. Example (Stewart, 14.4, Example 6) A box has dimensions 75 cm 60 cm 40 cm, each up to ±0.2 cm. What is (the magnitude of) the error bound on its volume?

49 If dx i is the error bound on a measurement of x i, then dz is the largest possible error in z resulting from measurements of x 1,..., x n. Example (Stewart, 14.4, Example 6) A box has dimensions 75 cm 60 cm 40 cm, each up to ±0.2 cm. What is (the magnitude of) the error bound on its volume? A box of dimensions r 1 r 2 r 3 has volume V = r 1 r 2 r 3. The differential is (18) dv = r 2 r 3 dr 1 + r 1 r 3 dr 2 + r 1 r 2 dr 3 and the answer is (19) (60)(40)(0.2) + (75)(40)(0.2) + (75)(60)(0.2) = 1980 cm 3.