Exercises VIII. (See page 257 for Answers.)

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

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

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

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

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

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