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
