MATH 180 FALL 1995 10:00 SECTION FONG

ASSIGNMENT #1

1.1, #4. The graph should have the following features: i) an initial increasing portion, increasing since “the more fertilizer is used, the better the yield”; ii) the starting value of the yield probably should be positive, since there is some yield even if no fertilizer is applied; iii) a final decreasing portion, decreasing since “if too much fertilizer is applied … the yield goes down rapidly”; iv) the decrease may be steep to indicate the yield descreases “rapidly”; v) whether the descending yield flattens out or goes to zero as the amount of fertilizer increases is open to interpretation.

1.1, #5. The graph says the productivity increases as the number of workers on the asembly line increases. But once past a certain number, productivity decreases as more workers are added to the assembly line. Possible reasons for the decrease?

1.1, #11. The domain of f consists of those portions of the x-axis lying under the graph, in this case all p such that −5 ≤ p ≤ 0 and all p such that 2 ≤ p < 6, i.e., the intervals [−5, 0] and [2, 6). The range of f consists of the values taken by f, in this case all r such that 0 ≤ r < ∞, i.e., the interval [0, ∞). Lastly, those r’s which occur as values exactly once correspond to points on the y-axis such that horizontal lines through them meet the graph exactly once, in this case all r such that 0 ≤ r < 2 and all r such that r > 5, i.e., the intervals [0, 2) and (5, ∞).

1.2, #1. The line given by 2y + 5x − 8 = 0 is the same as that given by y = −\frac{5}{2}x + 4. So the line has slope −\frac{5}{2} and y-intercept 4.

1.2, #2. The formula for a line on two given points works out to

\frac{y − 0}{x + 1} = \frac{6 − 0}{2 + 1} or y = 2x + 2.

1.2, #3. The formula for a line on a given point and parallel to a line with given slope works out to

\frac{y − c}{x − a} = m or y = mx + (c − ma).

Comments on #11: There were lots of answers which were wrong at endpoints of intervals, e.g., (0, ∞) instead of [0, ∞), [0, 2] instead of [0, 2). Be careful with such details. Some answers also did not express what the writers wanted to say. Here are some examples:

• Domain= −5 ≥ 0 and 2 ≥ all reals less than 6

• Range= 2 ≥ 5 and 0 ≥ ∞

• 2 > r > 5

None of these make mathematical sense. Learn the proper conventions for mathematical writing.