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

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

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;