CP-8 Hyperbolic functions — 9FM0 spec coverage

CP-8.1

Understand the definitions of hyperbolic functions sinh x, cosh x and tanh x, including their domains and ranges, and be able to sketch their graphs.

$\sinh x=\dfrac{e^x-e^{-x}}2$ , $\cosh x=\dfrac{e^x+e^{-x}}2$ and $\tanh x=\dfrac{\sinh x}{\cosh x}$ .
All three functions have domain $\mathbb R$ ; their ranges are $\mathbb R$ , $[1,\infty)$ and $(-1,1)$ respectively.
$\sinh x$ and $\tanh x$ are odd and increasing, while $\cosh x$ is even with minimum point $(0,1)$ ; $y=\pm1$ are asymptotes of $y=\tanh x$ .
Do not give $\tanh x$ the closed range $[-1,1]$ : its graph approaches but never reaches either horizontal asymptote.

Tier 1 · Easy

3 marks

ORIGINAL

State the domain, range and $y$ -intercept of $y=\cosh x$ .

Tier 2 · Standard

4 marks

ORIGINAL

Using the exponential definitions, find the exact values of $\sinh(\ln3)$ , $\cosh(\ln3)$ and $\tanh(\ln3)$ .

Tier 3 · Hard

5 marks

ORIGINAL

Sketch $y=2-3\tanh x$ . Label its intercept and both horizontal asymptotes, and state its range.

CP-8.2

Differentiate and integrate hyperbolic functions.

The basic derivatives are $(\sinh x)'=\cosh x$ , $(\cosh x)'=\sinh x$ and $(\tanh x)'=\operatorname{sech}^2x$ .
Apply the chain, product and quotient rules exactly as for other differentiable functions; the derivative of $\cosh x$ has no minus sign.
Reverse these results when integrating, including the inner derivative: for example $\int\sinh(ax+b)\,dx=\dfrac1a\cosh(ax+b)+C$ .
A common error is to copy circular-trigonometric signs, writing $(\cosh x)'=-\sinh x$ instead of $+\sinh x$ .

Tier 1 · Easy

2 marks

ORIGINAL

Differentiate $y=5\cosh(4x)$ with respect to $x$ .

Tier 2 · Standard

3 marks

ORIGINAL

Find $\displaystyle\int(5\cosh(3x)-2\sinh x)\,dx$ .

Tier 3 · Hard

6 marks

ORIGINAL

The function is $f(x)=e^{-x}\cosh(2x)$ . Find the exact coordinate of its stationary point and determine its nature.

CP-8.3

Understand and be able to use the definitions of the inverse hyperbolic functions and their domains and ranges.

$y=\operatorname{arsinh}x$ means $x=\sinh y$ ; it has domain $\mathbb R$ and range $\mathbb R$ .
$y=\operatorname{arcosh}x$ is the inverse of the restriction of $\cosh$ to $x\geq0$ , so its domain is $[1,\infty)$ and its range is $[0,\infty)$ .
$y=\operatorname{artanh}x$ means $x=\tanh y$ ; it has domain $(-1,1)$ and range $\mathbb R$ .
Do not treat $\cosh$ as one-to-one on all real numbers: its principal inverse returns the non-negative value only.

Tier 1 · Easy

2 marks

ORIGINAL

State the domain and range of $y=\operatorname{arcosh}x$ .

Tier 2 · Standard

3 marks

ORIGINAL

Solve $\operatorname{arsinh}(2x-1)=\ln2$ exactly.

Tier 3 · Hard

5 marks

ORIGINAL

Let $u=\operatorname{artanh}(1/3)$ . Find the exact value of $x$ satisfying $\operatorname{arcosh}x=2u$ , without using decimal approximations.

CP-8.4

Derive and use the logarithmic forms of the inverse hyperbolic functions.

$\operatorname{arsinh}x=\ln(x+\sqrt{x^2+1})$ for every real $x$ .
$\operatorname{arcosh}x=\ln(x+\sqrt{x^2-1})$ for $x\geq1$ , using the non-negative principal branch.
$\operatorname{artanh}x=\dfrac12\ln\left(\dfrac{1+x}{1-x}\right)$ for $|x|<1$ .
When deriving a logarithmic form, reject any algebraic root that would make $e^y\leq0$ , and retain the inverse function's domain restriction.

Tier 1 · Easy

2 marks

ORIGINAL

Use a logarithmic form to evaluate $\operatorname{arsinh}(3/4)$ exactly.

Tier 2 · Standard

4 marks

ORIGINAL

Solve $\operatorname{arsinh}x=\ln5-\ln2$ , giving $x$ exactly.

Tier 3 · Hard

6 marks

ORIGINAL

Using logarithmic forms, prove that $\operatorname{artanh}\left(\dfrac{x}{\sqrt{1+x^2}}\right)=\operatorname{arsinh}x$ for every real $x$ .

CP-8.5

Integrate functions of the form (x^2 + a^2)^(-1/2) and (x^2 - a^2)^(-1/2) and be able to choose substitutions to integrate associated functions.

For $\sqrt{x^2+a^2}$ , use $x=a\sinh u$ so that $\sqrt{x^2+a^2}=a\cosh u$ .
For $\sqrt{x^2-a^2}$ on $x\geq a>0$ , use $x=a\cosh u$ so that $\sqrt{x^2-a^2}=a\sinh u$ .
The standard antiderivatives are $\operatorname{arsinh}(x/a)+C$ for $(x^2+a^2)^{-1/2}$ and $\operatorname{arcosh}(x/a)+C$ for $(x^2-a^2)^{-1/2}$ on the stated branch.
Transform $dx$ and the limits as well as the radical; omitting one of these is the usual source of an incorrect scale factor.

Tier 1 · Easy

2 marks

ORIGINAL

Find $\displaystyle\int\dfrac{1}{\sqrt{x^2+16}}\,dx$ .

Tier 2 · Standard

6 marks

ORIGINAL

Using $x=3\sinh u$ , evaluate $\displaystyle\int_0^4\sqrt{x^2+9}\,dx$ exactly.

Tier 3 · Hard

6 marks

ORIGINAL

Use a hyperbolic substitution to evaluate $\displaystyle\int_3^5\dfrac{2x^2-9}{\sqrt{x^2-9}}\,dx$ exactly.