Sample Questions

Calculus I

Minima and Maxima

1) Find the critical points for the function f(X) = 4/3 * X3 + 4 * X2 + 3 * X - 12.

``` a)  X = 0.5 and X = 1.5.
```
``` b)  X = -0.5 and X = -1.5.
```
``` c)  X = 0.5 and X = -1.5.
```
``` d)  X = -0.5 and X = 1.5.
```
``` e)  None of the above.
```

2) Determine whether the extrema found in 1) are maxima or minima.

``` a)  X = -0.5: minimum; X = -1.5: maximum.
```
``` b)  X = -0.5: minimum; X = -1.5: minimum.
```
``` c)  X = -0.5: maximum; X = -1.5: maximum.
```
``` d)  X = -0.5: maximum; X = -1.5: minimum.
```
``` e)  None of the above.
```

3) Find the critical points for the fucntion f(X) = (1 - X) / (1 + X).

``` a)  X = 1.0.
```
``` b)  X = -1.0.
```
``` c)  X = 0.
```
``` d)  X = -2.0.
```
``` e)  None of the above.
```

4) Determine whether the extremum found in 3) is a maximum or minimum.

``` a)  minimum.
```
``` b)  maximum.
```
``` c)  optimum.
```
``` d)  point of inflection.
```
``` e)  None of the above.
```

5) A segment of a straight line starts at (0,5) and ends at (10,0). What are the coordinates of the point on the line segment that maximizes the area of the rectangle formed by the point, the origin, and the X and Y axes?

``` a)  P = (5.5, 2.0).
```
``` b)  P = (2.5, 5.0).
```
``` c)  P = (5.0, 2.5).
```
``` d)  P = (-5.0, -2.5).
```
``` e)  None of the above.
```

6) Confirm that the solution to 5) yields a maximum.

``` a)  f''(X) = 0.
```
``` b)  f''(X) = -1.0.
```
``` c)  f''(X) = 1.0.
```
``` d)  f''(X) = -12.5.
```
``` e)  None of the above.
```

7) A segment of the parabola f(X) = (10 - X)2 starts at (0,100) and ends at (0,10). What are the coordinates of the point on the arc segment that maximizes the area of the rectangle formed by the point, the origin, and the X and Y axes?

``` a)  P = (3.67, 40.11).
```
``` b)  P = (6.33, 40.11).
```
``` c)  P = (3.67, 6.33).
```
``` d)  P = (-3.67, 40.11).
```
``` e)  None of the above.
```

8) Confirm that the solution to 7) yields a maximum.

``` a)  f''(X) = 6 * X - 40.
```
``` b)  f''(X) = -18.0.
```
``` c)  f''(X) = 18.0.
```
``` d)  f''(X) = -22.0.
```
``` e)  None of the above.
```

9) A point Z is at (4, 8). What is the equation of the straight line, passing through the point Z, that minimizes the area of the triangle formed by the line and the X and Y axes? Hint: express the area as a function of the slope of the line.

``` a)  f(X) = 16 - X.
```
``` b)  f(X) = 8 - X.
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``` c)  f(X) = 16 - 2 * X.
```
``` d)  f(X) = 16 + 2 * X.
```
``` e)  None of the above.
```

10) Confirm that the solution to 9) yields a minimum.

``` a)  f''(X) = 16.0.
```
``` b)  f''(X) = -16.0.
```
``` c)  f''(X) = -8.0.
```
``` d)  f''(X) = 32 / m.
```
``` e)  None of the above.
```

11) A point Z is at (8,-6). What is the equation of the straight line, passing through the point Z, that minimizes the area of the triangle formed by the line and the X and Y axes? Hint: express the area as a function of the slope of the line.

``` a)  f(X) = 12 - 0.75 * X.
```
``` b)  f(X) = -12 + 0.75 * X.
```
``` c)  f(X) = -12 - 0.75 * X.
```
``` d)  f(X) = 12 + 0.75 * X.
```
``` e)  None of the above.
```

12) Confirm that the solution to 11) yields a minimum.

``` a)  f''(X) = 48.0.
```
``` b)  f''(X) = -0.75.
```
``` c)  f''(X) = 36.0.
```
``` d)  f''(X) = 85.3.
```
``` e)  None of the above.
```

13) A firm is hired to wire a lighthouse to an electrical station. The lighthouse is 15 kilometers off shore. The electrical station is located 20 kilometers downshore from the point directly accross from the lighthouse. It costs the firm \$100 per kilometer to lay wire under land and \$200 per kilometer to lay wire under water. What path should the firm take to minimize the cost of laying the wire? Hint: use the Pythagorean theorem.

``` a)  11.34 km under land and 17.32 km under water.
```
``` b)  17.32 km under land and 11.34 km under water.
```
``` c)  20.0 km under land and 15.0 km under water.
```
``` d)  8.66 km under land and 18.80 km under water.
```
``` e)  None of the above.
```

14) Confirm that the solution to 13) yields a minimum.

``` a)  f''(X) = -8.66.
```
``` b)  f''(X) = 11.55.
```
``` c)  f''(X) = 8.66.
```
``` d)  f''(X) = -2.88.
```
``` e)  None of the above.
```

15) A rectangular construction site needs to be fenced in and divided into 3 parts with additional fencing. The construction firm can purchase 1,200 meters of fencing. What are the dimensions of the largest possible area of the construction site?

``` a)  Width = 300 m and Length = 300 m.
```
``` b)  Width = 120 m and Length = 600 m.
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``` c)  Width = 300 m and Length = 150 m.
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``` d)  Width = 150 m and Length = 300 m.
```
``` e)  None of the above.
```

16) Confirm that the solution to 15) yields a maximum.

``` a)  f''(X) = -4 * W.
```
``` b)  f''(X) = 4.0.
```
``` c)  f''(X) = -4.0.
```
``` d)  f''(X) = -2.0.
```
``` e)  None of the above.
```

17) What are the dimensions of the isosceles triangle with a perimeter of 30 centimeters and with the largest possible area?

``` a)  Equilateral of side length 30 cm.
```
``` b)  Equilateral of side length 10 cm.
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``` c)  Isosceles of side length 12 cm and base length 6 cm.
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``` d)  A triangle of side lenths 8 cm, 10 cm, and 12 cm.
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``` e)  None of the above.
```

18) Confirm that the solution to 17) yields a maximum.

``` a)  f'(9) = 3.35 and f'(11) = -2.20 therefore maximum.
```
``` b)  f'(9) = -2.20 and f'(11) = 3.35 therefore maximum.
```
``` c)  f'(9) = 3.35 and f'(11) = -2.20 therefore minimum.
```
``` d)  f'(9) = 7.70 and f'(11) = -3.30 therefore maximum.
```
``` e)  None of the above.
```

19) What point on the parabola f(X) = 0.5 * X2 is closest to the point (0,4)?

``` a)  P = (3.0, 2.45).
```
``` b)  P = (6.0, 18.0).
```
``` c)  P = (1.22, 0.75).
```
``` d)  P = (2.45, 3.0).
```
``` e)  None of the above.
```

20) Confirm that the solution to 19) yields a minimum.

``` a)  f'(√5) = -0.415 and f'(√7) = 0.491 therefore minimum.
```
``` b)  f'(√5) = -0.415 and f'(√7) = 0.491 therefore maximum.
```
``` c)  f'(√5) = 0.415 and f'(√7) = -0.491 therefore minimum.
```
``` d)  f'(√5) = -0.491 and f'(√7) = 0.415 therefore maximum.
```
``` e)  None of the above.
```