Application of Differentiation

Report 1 / 5

Find the turning points of the curve

\(y=3x^3-5x^2+4\).

(A)		\(\begin{aligned} &(0,4) \\\\ &\text{and}\\\\ &\begin{pmatrix} \dfrac{10}{9},\dfrac{472}{243} \end{pmatrix} \end{aligned}\)

(B)		\(\begin{aligned} &(0,4) \\\\ &\text{and}\\\\ &\begin{pmatrix} \dfrac{10}{9},\dfrac{372}{243} \end{pmatrix} \end{aligned}\)

(C)		\(\begin{aligned} &(0,4) \\\\ &\text{and}\\\\ &\begin{pmatrix} \dfrac{10}{9},\dfrac{272}{243} \end{pmatrix} \end{aligned}\)

(D)		\(\begin{aligned} &(0,4) \\\\ &\text{and}\\\\ &\begin{pmatrix} \dfrac{10}{9},\dfrac{172}{243} \end{pmatrix} \end{aligned}\)

Difficulty Level: Hard

Report 2 / 5

Given that

\(y=x^3-9x^2-5\).

Find the turning points of the curve.

(A)		\(\begin{aligned} &(0,-5) \\\\ &\text{and}\\\\ &(6,-133) \end{aligned}\)

(B)		\(\begin{aligned} &(0,-5) \\\\ &\text{and}\\\\ &(6,-123) \end{aligned}\)

(C)		\(\begin{aligned} &(0,-5) \\\\ &\text{and}\\\\ &(6,-113) \end{aligned}\)

(D)		\(\begin{aligned} &(0,-5) \\\\ &\text{and}\\\\ &(6,-103) \end{aligned}\)

Difficulty Level: Medium

Report 3 / 5

The curve

\(y=ax^2+bx+c\)

has a turning point at \((-2,4)\) and passes through the point \((0,12)\).

Find the value of \(a\), of \(b\) and of \(c\).

(A)		\(\begin{aligned} a&=2 \\\\ b&=8 \\\\ c&=12 \end{aligned}\)

(B)		\(\begin{aligned} a&=2 \\\\ b&=8 \\\\ c&=10 \end{aligned}\)

(C)		\(\begin{aligned} a&=2 \\\\ b&=10 \\\\ c&=12 \end{aligned}\)

(D)		\(\begin{aligned} a&=2 \\\\ b&=10 \\\\ c&=8 \end{aligned}\)

Difficulty Level: Hard

Report 4 / 5

Given

\(y=3x^2-15x+7\),

find a small change for \(y\) when \(x\) increases from \(2\) to \(2.01\).

Difficulty Level: Medium

Report 5 / 5

Given that

\(y=\dfrac{32}{x^5}\),

find the value of \(\dfrac{dy}{dx}\) when \(x=2\).

Hence, estimate the value of \(\dfrac{32}{2.02^5}\).

Difficulty Level: Hard