Chapter Questions
Differentiate the following functions(i) $y=\frac{1}{x^3}$(ii) $y=\sqrt[3]{x}$(iii) $y=\frac{2}{x}-\frac{3}{x^2}$(iv) $y=4 \sqrt{x}-\frac{3}{\sqrt{x}}$(v) $y=3 x^{-5}-2 x^{-7}$(vi) $y=2 x^{\frac{2}{3}}-5 x^{-\frac{2}{3}}$(vii) $y=\frac{x^2-2 x+3}{2 x^2}$(viii) $y=\left(x^2-2\right) \sqrt{x}$
The derivative of $\sqrt[4]{x}$ is(a) $-4 x^{-5}$(b) $\frac{1}{4} x^{-\frac{1}{4}}$(c) $\frac{1}{4} x^{-\frac{1}{4}}$(d) $-4 x^{-3}$(e) I don't know
Find the gradient of each of the following graphs at the given point(i) $y=2 x-\frac{1}{x}$ at the point $(1,1)$(ii) $y=3-\sqrt{x}$ at the point $(4,1)$(iii) $y=x^2 \sqrt{x}$ at the point $(1,1)$
The derivative of $\frac{1}{x^5}$ is(a) $-\frac{5}{x^4}$(b) $-\frac{5}{x^6}$(c) $\frac{5}{x^4}$(d) $\frac{1}{5 x^4}$(e) I don't know
Find any stationary points on the following curves and determine their nature.(i) $y=x-\frac{4}{x^2}$(ii) $y=\sqrt{x}+\frac{1}{\sqrt{x}}$
The derivative of $(3 x-2) \sqrt{x}$ is(a) $3 \sqrt{x}-\frac{2}{\sqrt{x}}$(b) $\frac{3}{2 \sqrt{x}}$(c) $\frac{3 x-2}{2 \sqrt{x}}$(d) $\frac{9 \sqrt{x}}{2}-\frac{1}{\sqrt{x}}$(e) I don't know
Find the equation of the tangent to the graph $y=\frac{1}{\sqrt{x}}$ at the point where $x=1$.
The derivative of $\frac{x^4-2 x^2+3}{x^3}$ is(a) $-1+\frac{3}{x^2}$(b) $\frac{4 x^3-4 x}{3 x^2}$(c) $x-\frac{2}{x}+\frac{3}{x^3}$(d) $1+\frac{2}{x^2}-\frac{9}{x^4}$(e) I don't know
Find the equation of the normal to the graph $y=\frac{1}{x}-\frac{2}{x^2}$ at the point where $x=2$.
The gradient of the curve $y=\frac{2}{\sqrt{x}}$ at the point $(4,1)$ is(a) $-\frac{1}{8}$(b) $\frac{1}{8}$(c) $\frac{1}{2}$(d) $-\frac{1}{2}$(e) I don't know
A cylindrical can with height $h$ metres and radius $r$ metres has a capacity of 2 litres.(i) Find an expression for $h$ in terms of $r$.(ii) Hence find an expression for the surface area of the can in terms of $r$ only.(iii) Find the value of $r$ which minimises the surface area of the can.
The gradient of the curve $y=\frac{2}{x}-\frac{3}{x^2}$ at the point ( $2, \frac{1}{4}$ ) is(a) $\frac{1}{2}$(b) $-\frac{1}{4}$(c) $\frac{1}{4}$(d) $\frac{5}{4}$(e) I don't know
The equation of the tangent to the curve $y=2 x-\frac{1}{x}$ at the point where $x=1$ is(a) $y=3 x$(b) $y=3 x-2$(c) $y=x$(d) $y=x-1$(e) I don't know
The equation of the normal to the curve $y=2 x-\sqrt{x}$ at the point where $x=4$ is(a) $7 y=4 x+26$(b) $4 y=7 x+8$(c) $7 y+4 x=58$(d) $4 y+7 x=52$(e) I don't know
The stationary point of the curve $y=x^2-\frac{2}{x}$ is(a) Maximum $(-1,3)$(b) Minimum $(-1,3)$(c) Maximum $(1,-1)$(d) Minimum $(1,-1)$(e) I don't know
The stationary point of the curve $y=x-4 \sqrt{x}$ is(a) Minimum $(4,-4)$(b) Maximum $(4,-4)$(c) Minimum $\left(\frac{1}{4},-\frac{7}{4}\right)$(d) Maximum $\left(\frac{1}{4},-\frac{7}{4}\right)$(e) I don't know