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Math 281 Intermediate Calculus

Math 281 Intermediate Calculus. In this course, we will study Calculus of functions of several variables and Analytic geometry of 3 dimensional objects. What can Calculus do?. In Calculus I, we saw that differentiation can be used to find the slope of a curve at any (smooth) point.

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Math 281 Intermediate Calculus

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  1. Math 281Intermediate Calculus • In this course, we will study • Calculus of functions of several variables and • Analytic geometry of 3 dimensional objects.

  2. What can Calculus do? In Calculus I, we saw that differentiation can be used to find the slope of a curve at any (smooth) point.

  3. We can also use differentiation to find the local max and local minimum of a function.

  4. In this course, we will learn how to find the slope of a surface at any smooth point, and in any direction.

  5. Absolute max local max absolute min local min We will also learn how to find the local maximum and local minimum of a surface.

  6. In Calculus I, we saw that Integration can be used to find the area between a curve and the x-axis (from x = a to x = b )

  7. In this course, we will learn how to find the volume of the region bounded by several surfaces.

  8. z 1 0.5 0 -0.5 -1 -1 -1 -0.5 -0.5 0 x 0 0.5 y 0.5 1 1 And the volumes of many other solids, such as a bumpy sphere!

  9. Of course, you will need to learn many other useful mathematical techniques in this program. Therefore you need to work hard and learn fast. Plan to study at least 4 hours each day.

  10. Review Recall that for functions of one variable such as f(x) = 2exsinx ,we can draw its graph on a piece of paper because we need only two perpendicular axes, one for the input (i.e. the x value) and one for the output (i.e. the y value) y x

  11. However, for functions of two variables, such as f (x ,y) = 2x2y + xy2 the graph will no longer be a curve and we need 3 mutually perpendicular axes to represent the two inputs and one output. The output value is usually denoted by the letter z. i.e. z = 2x2y + xy2 In order to accommodate the z axis that is perpendicular to both the x and y axes, we need to make it coming out from the screen (see the next slide).

  12. Since the z axis is coming out from the xy-plane, we cannot literally draw this on a piece of paper. You need to use a bit of imagination to see that the z axis is really coming out. We also need to rotate the axes so that the z axis is pointing up, because we are used to seeing the output on the vertical axis.

  13. The coordinates of any point in space will be an ordered triple (x, y, z). The last component z will tell us how far the point is above or below the horizontal plane. z This will then be the standard orientation of the 3 axes, and most of our diagrams will be drawn this way. y x

  14. z (1) the xy-plane In addition to the 3 coordinate axes, we also have 3 coordinate planes namely y x

  15. z (1) the xy-plane In addition to the 3 coordinate axes, we also have 3 coordinate planes namely (2) the xz-plane y x

  16. z (1) the xy-plane In addition to the 3 coordinate axes, we also have 3 coordinate planes namely (2) the xz-plane (3) the yz-plane y x x

  17. The 1st octant is the one where all coordinates are positive. z These 3 coordinate planes will divide the space into 8 congruent pieces, each one is called an octant. y x The other octants are not labelled.

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