Maths4U Original · Short practice

Let \(f(x)=5 x^{2} e^{x}\), with x>0.

Find f′(1).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(f(x)=2 x^{3} \log{\left(x \right)}\), with x>0.

Find f′(1).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The curve satisfies \(x^2+3xy+y^2=31\). Consider \(\displaystyle P=\)\(\left( 3, \ 2\right)\). For the short checks \(\displaystyle a=3\).

Find the tangent y-intercept there.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The curve satisfies \(x^2+1xy+y^2=37\). Consider \(\displaystyle P=\)\(\left( 4, \ 3\right)\). For the short checks \(\displaystyle a=1\).

Find the tangent y-intercept there.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

A curve has \(x=t^2+3\), \(y=t^3+2t\). Consider \(\displaystyle t=3\).

Find \(\displaystyle \frac{dy}{dx}\) there.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

A curve has \(x=t^2+4\), \(y=t^3+3t\). Consider \(\displaystyle t=1\).

Find \(\displaystyle \frac{dy}{dx}\) there.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(f(x)=19 x + 4 \operatorname{atan}{\left(x \right)}\), with real x.

Find the limit of f′(x) as x tends to positive infinity.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

Let \(f(x)=20 x + 5 \operatorname{atan}{\left(2 x \right)}\), with real x.

Find the limit of f′(x) as x tends to positive infinity.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.