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



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.

10. Find the slope of the tangent to the curve $$y = 2x^3 - 6x + 5$$, (a) at the point where $$x = 1$$; (b) at the point where $$x = 0$$. Ans. (a) 0; (b) -6.

11. (a) Find the slopes of the tangents to the two curves $$y = 3x^2 - 1$$ and $$y = 2x^2 + 3$$ at their points of intersection. (b) At what angle do they intersect?

Ans. (a) $\pm 12, \pm 8$; (b) $\arctan \frac{4}{97}$.

12. The curves on a railway track are often made parabolic in form. Suppose that a track has the form of the parabola $$y = x^2$$ (last figure, § 32), the directions $$OX$$ and $$OY$$ being east and north respectively, and the unit of measurement 1 mile. If the train is going east when passing through $$O$$, in what direction will it be going

13. A street-car track has the form of the cubical parabola $$y = x^3$$. Assume the same directions and unit as in the last example. If a car is going west when passing through $$O$$, in what direction will it be going