There are many functions that you should have on your fingertips,
in the following sense:
First review:
Basic Models You Must Know
The current lesson builds on this prior information.
Most graphs are shown in the same window,
from $\,-5\,$ to $\,5\,$ on the x-axis,
and from $\,-5\,$ to $\,5\,$ on the y-axis.
You may use the navigation arrows (bottom right on each graph) to zoom or move around.
Drag the point on each graph to see coordinates.
The scales on the x-axis and y-axis are identical.
That is, a distance of $\,1\,$ on the x-axis is the same as a distance of $\,1\,$ on the y-axis.
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the natural exponential function $f(x) = {\text{e}}^x$ or $f(x) = \exp(x)$ other bases: $y = 2^x$ $y = 3^x$ $y = (\frac{1}{2})^x$ detail |
There is an entire family of
exponential functions;
they are of the form $\,y = b^x\,$, where $\,b\,$ is a positive number, not equal to $\,1\,$. For example, $\ y = 2^x\ $ and $\ y = (\frac{1}{2}){}^x\ $ are exponential functions. Thus, exponential functions have a constant base; the variable is in the exponent. Of all the exponential functions, $\,f(x) = {\text{e}}^x\,$ (base $\,\text{e}\,$) is singled out as most important, due to the simplicity of a calculus property it possesses:
the slope of the tangent line at the point $\,(x,{\text{e}}^x)\,$ is $\,m = {\text{e}}^x\,$;
Think about this!that is, the $y$-value of the point gives the slope of the tangent line If $y = 100\,$, then the slope of the tangent line there is $\,100\,$, so the $y$-values are increasing $\,100\,$ times faster than the inputs! (Exponential functions with other bases have a similar, but not-so-simple property.)
Exponential functions get big FAST!
$f(x) = {\text{e}}^x\,$ is given the special name ‘the natural exponential function’
and is also denoted by $\,f(x) = \exp(x)\,$.Recall that $\,\text{e}\,$ is defined as the number that $\,(1 + \frac 1n)^n\,$ approaches as $n\rightarrow\infty\,$; $\,\text{e}\,$ is irrational; $\,\text{e}\approx 2.71828\,$. If you hear the phrase ‘the exponential function’ (meaning only one) then the function being referred to is the natural exponential function. Properties that ALL exponential functions share:
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the natural logarithm function $f(x) = \ln(x)$ other bases: $y = \log_2(x)$ $y = \log_3(x)$ $y = \log_{1/2}(x)$ detail |
There is an entire family of
logarithmic functions;
they are of the form $\,y = \log_b(x)\,$, where $\,b\,$ is a positive number, not equal to $\,1\,$. For example, $\ y = \log_2(x)\ $ and $\ y = \log_{1/2}(x)\ $ are logarithmic functions. When the base is the irrational number $\,\text{e}\,$ the function is given a special name, $\,f(x) = \ln(x) := \log_{\text{e}}(x)\,$. Of all the logarithmic functions, $\,f(x) = \ln(x)\,$ is singled out as most important, due to the simplicity of a calculus property it possesses:
the slope of the tangent line at the point $\,(x,\ln x)\,$ is $\,m = \frac 1x\,$;
Think about this!that is, the reciprocal of the $x$-value of the point gives the slope of the tangent line If $x = 100\,$, then the slope of the tangent line there is only $\,\frac{1}{100}\,$, so the $y$-values are increasing only $\,\frac{1}{100}\,$ times as fast as the inputs! (Logarithmic functions with other bases have a similar, but not-so-simple property.)
Logarithmic functions get big, but they get big VERY SLOWLY!
$f(x) = \ln(x)\,$ is given the special
name ‘the natural logarithm function’.
Recall that $\,\text{e}\,$ is defined as the number that $\,(1 + \frac 1n)^n\,$ approaches as $n\rightarrow\infty\,$; $\,\text{e}\,$ is irrational; $\,\text{e}\approx 2.71828\,$. Some disciplines use $\,\log(x)\,$ (no indicated base) to mean the natural logarithm; some disciplines use $\,\log(x)\,$ to mean the common logarithm (base ten). You'll need to check notation. Properties that ALL logarithmic functions share:
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![]() the sine function $f(x) = \sin(x)$ the cosine function $f(x) = \cos(x)$ |
The sine function gives the $y$-values of points on the unit circle. The cosine function gives the $x$-values of points on the unit circle. (See the discussion below to understand how to associate real numbers, $\,x\,$, with points on the unit circle.) Both sine and cosine share the following properties:
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the tangent function $f(x) = \tan(x)$ |
The tangent function is defined as:
$$\cssId{sb119}{\tan(x) := \frac{\sin(x)}{\cos(x)}}$$
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On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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