5441794
Beschreibung
Karteikarten von Daniel Cox, aktualisiert more than 1 year ago
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Erstellt von Daniel Cox
vor mehr als 8 Jahre
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| Frage | Antworten |
| \[\sec x\equiv ?\] | \[\sec x\equiv \frac{1}{\cos x}\] |
| \[\text{cosec } x\equiv ?\] | \[\text{cosec }x\equiv \frac{1}{\sin x}\] |
| \[\cot x\equiv ?\] | \[\cot x\equiv \frac{1}{\tan x}\] |
| Write an identity which links together \(\sec \theta\) and \(\tan \theta\) | \[\sec^2 \theta \equiv 1 + \tan^2 \theta\] |
| Write an identity which links together \(\text{cosec }\theta\) and \(\cot \theta\) | \[\text{cosec}^2 \theta \equiv 1 + \cot^2 \theta\] |
| Sketch the graph of \(y=\sin x\) for \(0\leq x \leq 360^{\circ}\) | |
| Sketch the graph of \(y=\cos x\) for \(0\leq x \leq 360^{\circ}\) | |
| Sketch the graph of \(y=\tan x\) for \(0\leq x \leq 360^{\circ}\) | |
| Sketch the graph of \(y=\sin^{-1}x\). What is its domain and range? | Domain \([-1,1]\) Range \([-\frac{\pi}{2},\frac{\pi}{2}]\) |
| Sketch the graph of \(y=\cos^{-1}x\). What is its domain and range? | Domain \([-1,1]\) Range \([0,\pi]\) |
| Sketch the graph of \(y=\tan^{-1}x\). What is its domain and range? | Domain \((-\infty,\infty)\) Range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) |
| What does \(|x|<a\) mean? | \[-a<x<a\] \[\text{NOT } x<\pm a\] |
| What does \(|x|>a\) mean? | \[x>a \text{ or } x<-a\] \[\text{NOT } x>\pm a\] |
| Give the definition of a function | A function relates each element of a set with exactly one element of another set |
| How are the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\) related? | They are reflections of each other in the line \(y=x\) |
| The domain of \(f(x)\) equals the range of ...? | The domain of \(f(x)\) equals the range of \(f^{-1}(x)\) |
| The range of \(f(x)\) equals the domain of ...? | The range of \(f(x)\) equals the domain of \(f^{-1}(x)\) |
| What is meant by the range of a function? | The set of all possible output values of a function |
| What is meant by the domain of a function? | All the values that could go into a function |
| State the inverse function of \(e^x\) | \[\ln x\] |
| Sketch the graph of \(y=e^x\), showing any intersections with the axes. State its domain and range. | The \(x\) axis is an asymptote. Domain \((-\infty, \infty)\) Range \((0, \infty)\) |
| Sketch the graph of \(y=\ln x\), showing any intersections with the axes. State its domain and range. | The \(y\) axis is an asymptote. Domain \((0, \infty)\) Range \((-\infty, \infty)\) |
| State the inverse function of \(\ln x\) | \[e^x\] |
| Differentiate \(e^{f(x)}\) with respect to \(x\) | \[f'(x)e^{f(x)}\] |
| Differentiate \(\ln{f(x)}\) with respect to \(x\) | \[\frac{f'(x)}{f(x)}\] |
| Differentiate \(\sin {\left(f(x)\right)}\) with respect to \(x\) | \[f'(x)\cos{\left(f(x)\right)}\] |
| Differentiate \(\cos {\left(f(x)\right)}\) with respect to \(x\) | \[-f'(x)\sin{\left(f(x)\right)}\] |
| Differentiate \(\tan {\left(f(x)\right)}\) with respect to \(x\) | \[f'(x)\sec^2{\left(f(x)\right)}\] |
| State the product rule for differentiating \[y=uv\] with respect to \(x\), where \(u\) and \(v\) are functions of \(x\) | \[\frac{\text{d}y}{\text{d}x}=uv'+vu'\] |
| State the quotient rule for differentiating \[y=\frac{u}{v}\] with respect to \(x\), where \(u\) and \(v\) are functions of \(x\) | \[\frac{\text{d}y}{\text{d}x}=\frac{vu'-uv'}{v^2}\] |
| \[\frac{1}{\left(\frac{\text{d}x}{\text{d}y}\right)}=?\] | \[\frac{\text{d}y}{\text{d}x}\] |
| How would you find the first and second derivatives of the parametric equations \(x=f(t)\) and \(y=g(t)\) | \[\frac{\text{d}y}{\text{d}x}=\frac{g'(t)}{f'(t)}\] \[\frac{\text{d}^2y}{\text{d}x^2}=\frac{\frac{\text{d}}{\text{d}t}\left(\frac{\text{d}y}{\text{d}x}\right)}{\frac{\text{d}x}{\text{d}t}}\] |
| \[\int e^x \, dx =?\] | \[\int e^x \, dx = e^x+c\] |
| \[\int \frac{1}{x} \, dx = ?\] | \[\int \frac{1}{x} \, dx = \ln x +c\] |
| \[\int \sin x \, dx = ?\] | \[\int \sin x \, dx = -\cos x +c\] |
| \[\int \cos x \, dx = ?\] | \[\int \cos x \, dx = \sin x +c\] |
| What effect will the transformation \(y=f(x)+a\) have on the graph of \(y=f(x)\)? | Translation \(a\) units in the \(y\) direction. i.e. the graph will move UP by \(a\) units |
| What effect will the transformation \(y=f(x+a)\) have on the graph of \(y=f(x)\)? | Translation \(-a\) units in the \(x\) direction. i.e. the graph will move LEFT by \(a\) units |
| What effect will the transformation \(y=af(x)\) have on the graph of \(y=f(x)\)? | Stretch, scale factor \(a\) in the \(y\) direction. i.e. the \(y\) values will be multiplied by \(a\) |
| What effect will the transformation \(y=f(ax)\) have on the graph of \(y=f(x)\)? | Stretch, scale factor \(\frac{1}{a}\) in the \(x\) direction. i.e. the \(x\) values will be divided by \(a\) [This could also be described as a 'squash', scale factor \(a\) in the \(x\) direction] |
| Sketch the graph of \[y=|x|\] |
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