Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/57

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EXAMPLES

Find by differentiation the slopes of the tangents to the following curves at the points indicated. Verify each result by drawing the curve and its tangent.

1. , where . Ans. 4.
2. where . -6.
3. , where . -3.
4. , where . .
5. , where . 1.
6. , where . .
7. , where . 4.
8. , where . 0.
9. , where . 6.

10. Find the slope of the tangent to the curve , (a) at the point where ; (b) at the point where .

Ans. (a) 0; (b) -6.

11. (a) Find the slopes of the tangents to the two curves and at their points of intersection. (b) At what angle do they intersect?

Ans. (a) ; (b) .

12. The curves on a railway track are often made parabolic in form. Suppose that a track has the form of the parabola (last figure, § 32), the directions and being east and north respectively, and the unit of measurement 1 mile. If the train is going east when passing through , in what direction will it be going

(a) when mi. east of ? Ans. Northeast.
(b) when mi. west of ? Southeast.
(c) when mi. east of ? N. 30°E.
(d) when mi. north of ? E. 30°S., or E. 30°N.

13. A street-car track has the form of the cubical parabola . Assume the same directions and unit as in the last example. If a car is going west when passing through , in what direction will it be going

(a) when mi. east of ? Ans. Southwest.
(b) when mi. west of ? Southwest.
(c) when mi. north of ? S. 27° 43' W.
(d) when 2 mi. south of ?
(e) when equidistant from and ?