Page:Calculus Made Easy.pdf/108

The following examples show further applications of the principles just explained.

(4) Find the slope of the tangent to the curve

$y={\frac {1}{2x}}+3$,

at the point where $x=-1$. Find the angle which this tangent makes with the curve $y=2x^{2}+2$.

The slope of the tangent is the slope of the curve at the point where they touch one another (see p.77); that is, it is the ${\dfrac {dy}{dx}}$ of the curve for that point. Here ${\dfrac {dy}{dx}}=-{\dfrac {1}{2x^{2}}}$ and for ${\dfrac {dy}{dx}}=-{\dfrac {1}{2}}$, which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation $y=ax+b$, and its slope is ${\dfrac {dy}{dx}}=a$, hence $a=-{\dfrac {1}{2}}$. Also if $x=-1$, $y={\dfrac {1}{2(-1)}}+3=2{\frac {1}{2}}$; and as the tangent passes by this point, the coordinates of the point must satisfy the equation of the tangent, namely

$y=-{\dfrac {1}{2}}x+b$,

so that $2{\frac {1}{2}}=-{\dfrac {1}{2}}\times (-1)+b$ and $b=2$; the equation of the tangent is therefore $y=-12x+2$.

Now, when two curves meet, the intersection being a point common to both curves, its coordinates must satisfy the equation of each one of the two curves;