Page:Calculus Made Easy.pdf/111

 Exercises VIII. (See page 257 for Answers.)

(1) Plot the curve $$y=\tfrac{3}{4}x^2-5$$, using a scale of millimetres. Measure at points corresponding to different values of $$x$$, the angle of its slope.

Find, by differentiating the equation, the expression for slope; and see, from a Table of Natural Tangents, whether this agrees with the measured angle.

(2) Find what will be the slope of the curve

at the particular point that has as abscissa $$x=2$$.

(3) If $$y=(x-a)(x-b)$$, show that at the particular point of the curve where $$\dfrac{dy}{dx}=0$$, $$x$$ will have the value $$\tfrac{1}{2}(a+b)$$.

(4) Find the $$\dfrac{dy}{dx}$$ of the equation $$y = x^3 + 3x$$; and calculate the numerical values of $$\dfrac{dy}{dx}$$ for the points corresponding to $$x=0$$, $$x=\tfrac{1}{2}$$, $$x=1$$, $$x=2$$.

(5) In the curve to which the equation is $$x^2+y^2=4$$, find the values of $$x$$ at those points where the slope = $$1$$.

(6) Find the slope, at any point, of the curve whose equation is $$\dfrac{x^2}{3^2}+\dfrac{y^2}{2^2}=1$$; and give the numerical value of the slope at the place where $$x=0$$, and at that where $$x=1$$.

(7) The equation of a tangent to the curve $$y=5-2x+0.5x^3$$, being of the form $$y=mx+n$$, where m and n are constants, find the value of $$m$$ and $$n$$ if