By Robert Miller
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Extra resources for Calc I
Example 1— y = 0 means the top is 0. ''Top is 0" means x = 0, 2x - 3 = 0, or x + 4 = 0. You ignore the exponents, since x4 = 0 means x = 0. x = 0, 3/2, and -4. 5,0), and (-4,0). Example 2— Factor the top. (x-3)(x+1)=0. Intercepts are (3,0) and (-1,0). Example 3— Intercepts are (0,0), (1,0), and (2,0). y intercepts. Just like the line, y intercept means a point where x = 0. Example 4— Substitute x=0. y intercept is (0,8). Example 5— For x = 0, we get -3/0. There is no y intercept. Warnings: 1. If you get the sign of the y intercept wrong, you will never, never sketch the curve properly.
Note 2 At most there is one oblique asymptote or one horizontal asymptote, but not both. There might be neither. Curve Sketching By the Pieces Before we take a long example, we will examine each piece. When you understand each piece, the whole will be easy. Example 11— The intercept is (4,0). We would like to know what the curve looks like near (4,0). Except at the point (4,0), we do not care what the exact value is for y, which is necessary in an exact graph. In a sketch we are only interested in the sign of the y values.
N). C. Looking at the (∆x) 2 terms, we would get Factoring, we would get 6. Now, adding everything up, multiplying by ∆x, hoping everything fits on one line, we get 7. Substituting the formulas in the beginning and remembering ∆x = 3/n, we get 8. Using the distributive law, we get three terms: A. B. C. 9. Adding A + B + C, we get a value for the area of 84 + 45 + 9 = 138. Note By letting n go to infinity, we are doing two things: chopping up the interval 3 < x < 6 into more and more rectangles and, since ∆x = 3/n, making each rectangle narrower and narrower.