Maths4U Original · Short practice

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

Find f′(0).

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 \(f(x)=5 \sin{\left(2 x \right)} + 20 \cos{\left(2 x \right)}\), with real x.

Find f′(0).

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 \(f(x)=5 \sin{\left(2 x \right)} + 20 \cos{\left(2 x \right)}\), with real x.

Find f″(0).

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 \(f(x)=2 \sin{\left(3 x \right)} + 21 \cos{\left(3 x \right)}\), with real x.

Find f″(0).

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 \(f(x)=21 x + 2 \tan{\left(3 x \right)}\), with \(\displaystyle x\ne \frac{\frac{\pi }{2}+ n\pi }{3}, \quad n\in \mathbb{Z}\).

At \(\displaystyle x= \frac{\pi }{12}\), write \(\displaystyle f(x)=a+b\pi\) with rational a,b. Find b.

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 \(f(x)=22 x + 3 \tan{\left(x \right)}\), with \(\displaystyle x\ne \frac{\frac{\pi }{2}+ n\pi }{1}, \quad n\in \mathbb{Z}\).

At \(\displaystyle x= \frac{\pi }{4}\), write \(\displaystyle f(x)=a+b\pi\) with rational a,b. Find b.

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 \(f(x)=22 x + 3 \tan{\left(x \right)}\), with \(\displaystyle x\ne \frac{\frac{\pi }{2}+ n\pi }{1}, \quad n\in \mathbb{Z}\).

Find the spacing between successive vertical asymptotes, divided by π.

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 \(f(x)=23 x + 4 \tan{\left(2 x \right)}\), with \(\displaystyle x\ne \frac{\frac{\pi }{2}+ n\pi }{2}, \quad n\in \mathbb{Z}\).

Find the spacing between successive vertical asymptotes, divided by π.

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 f′(0).

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 \(f(x)=21 x + 2 \operatorname{atan}{\left(3 x \right)}\), with real x.

Find f′(0).

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 \(f(x)=21 x + 2 \operatorname{atan}{\left(3 x \right)}\), with real x.

Find \(\displaystyle f'(\frac{1}{3})\).

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 \(f(x)=22 x + 3 \operatorname{atan}{\left(x \right)}\), with real x.

Find \(\displaystyle f'(\frac{1}{1})\).

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 \(f(x)=22 x + 3 \operatorname{atan}{\left(x \right)}\), with real x.

Find f″\(\displaystyle (\frac{1}{1})\).

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 \(f(x)=23 x + 4 \operatorname{atan}{\left(2 x \right)}\), with real x.

Find f″\(\displaystyle (\frac{1}{2})\).

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 \(f(x)=23 x + 4 \operatorname{atan}{\left(2 x \right)}\), 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.

Maths4U Original · Short practice

Let \(f(x)=24 x + 5 \operatorname{atan}{\left(3 x \right)}\), 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.