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.

1. $y=x^{2}-4$,	where x = 2.	Ans. 4.
2. $y=6-3x^{2}$	where x = 1.	-6.
3. $y=x^{3}$,	where x = -1.	-3.
4. $y={\frac {2}{x}}$,	where x = -1.	$-{\frac {1}{2}}$.
5. $y=x-x^{2}$,	where x = 0.	1.
6. $y={\frac {1}{x-1}}$,	where x = 3.	$-{\frac {1}{4}}$.
7. $y={\frac {1}{2}}x^{2}$,	where x = 4.	4.
8. $y=x^{2}-2x+3$,	where x = 1.	0.
9. $y=9-x^{2}$,	where x = -3.	6.

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

(a) when ${\tfrac {1}{2}}$ mi. east of $OY$?	Ans. Northeast.
(b) when ${\tfrac {1}{2}}$ mi. west of $OY$?	Southeast.
(c) when ${\tfrac {\sqrt {3}}{2}}$ mi. east of $OY$?	N. 30°E.
(d) when ${\tfrac {1}{12}}$ mi. north of $OX$?	E. 30°S., or E. 30°N.

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

(a) when ${\tfrac {1}{\sqrt {3}}}$ mi. east of $OY$?	Ans. Southwest.
(b) when ${\tfrac {1}{\sqrt {3}}}$ mi. west of $OY$?	Southwest.
(c) when ${\tfrac {1}{2}}$ mi. north of $OX$?	S. 27° 43' W.
(d) when 2 mi. south of $OX$?
(e) when equidistant from $OX$ and $OY$?

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