![](\"resources/question-resources/sinh_sketch.png\"/)
Quiz designed as summary test for MAST resource: Hyperbolic Functions (A Level Further Maths content)
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", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "Which hyperbolic function is represented by this graph?
", "advice": "This is a graph of
\n$y=\\sinh(x)=\\frac{e^{x}-e^{-x}}{2}$
\nNotice that at $x$ increases, $\\sinh(x)$ looks like an exponential curve. This is because it approximates $\\frac{e^x}{2}$ as the $e^{-x}$ term becomes negligible.
\nSimilarly, as $x$ decreases, the graph looks likle the curve $-\\frac{e^{-x}}{2}$.
\n$y=\\cosh(x)=\\frac{e^{x}+e^{-x}}{2}$
\nNotice that at $x$ increases, $\\cosh(x)$ looks like an exponential curve. This is because it approximates $\\frac{e^x}{2}$ as the $e^{-x}$ term becomes negligible.
\nSimilarly, as $x$ decreases, the graph looks like the curve $\\frac{e^{-x}}{2}$.
\nYou may recall that this curve can be though of as an average of the functions $\\frac{e^x}{2}$ and $\\frac{e^{-x}}{2}$
\n$y=\\tanh(x)=\\frac{\\sinh(x)}{\\cosh(x)}=\\frac{e^{2x}-1}{e^{2x}+1}$
\nNotice that at $x$ increases in the positive direction, $\\tanh(x)$ tends to 1. Similarly, as $x$ increases in the negative directions, $\\tanh(x) \\rightarrow -1$.
\nYou may wish to look back at the graphs chapter of this resource for a more thorough explanation.
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![](\"resources/question-resources/cosh_sketch.png\"/)
![](\"resources/question-resources/tanh_sketch.png\"/)
Differentiate the following functions: $\\displaystyle x ^ n \\sinh(ax + b),\\;\\tanh(cx+d),\\;\\ln(\\cosh(px+q))$
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Write down the derivatives of the following functions $f(x)$ .
\nNote that in order to input the square of a function such as $\\sinh(x)$ you have to input it as sinh(x)^2, similarly for the other hyperbolic functions.
Here is a table of the derivatives of some of the hyperbolic functions:
\n| $f(x)$ | $\\displaystyle{\\frac{df}{dx}}$ |
|---|---|
| $\\sinh(bx)$ | \n$b\\cosh(bx)$ | \n
| $\\cosh(bx)$ | \n$b\\sinh(bx)$ | \n
| $\\tanh(bx)$ | \n$\\simplify{b*sech(bx)^2}$ | \n
$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$
\nUse the product rule to obtain:
\\[\\frac{df}{dx} = \\simplify[std]{{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})}\\]
$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$
\nUsing the table above we get:
\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]
$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$
\nUsing the chain rule we find:
\n\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]
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\n$\\displaystyle{\\frac{dy}{dx}=\\;\\;}$[[0]]
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\n$\\displaystyle{\\frac{dy}{dx}=\\;\\;}$[[0]]
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\n$\\displaystyle{\\frac{dy}{dx}=\\;\\;}$[[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n \n \nIntegrate the following functions $f(x)$.
\n \n \n ", "advice": "\\[\\int \\simplify[std]{cosh({a}x+{b})}\\;dx = \\frac{1}{\\var{a}}\\simplify[std]{ sinh({a}x+{b})}+C\\]
\nIntegrate by parts, so that
\n\\[\\begin{eqnarray*} \\int \\simplify[std]{x*sinh({a1}x+{b1})}\\;dx&=&\\int \\simplify[std]{x*d((cosh({a1}x+{b1}))/{a1})}\\\\ &=&\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1}} \\int \\simplify[std]{cosh({a1}x+{b1})}\\;dx\\\\ &=&\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1^2}}\\simplify[std]{sinh({a1}x+{b1})}+C \\end{eqnarray*} \\]
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\n$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]
\nYou must include the constant of integration as $C$.
\nInput all numbers as integers or fractions – not as decimals.
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\n$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]
\nInclude the constant of integration as $C$.
\nInput all numbers as integers or fractions – not as decimals.
\nPlease note that if you want to enter a function of the form $xf(x)$ then enter as x*f(x) so that it's clear what you mean.
Input all numbers as integers or fractions – not as decimals.
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