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 xaxis,
and from $\,5\,$ to $\,5\,$ on the yaxis.
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 xaxis and yaxis are identical.
That is, a distance of $\,1\,$ on the xaxis is the same as a distance of $\,1\,$ on the yaxis.















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 notsosimple 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:


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 notsosimple 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:


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:


the tangent function $f(x) = \tan(x)$ 
The tangent function is defined as:
$$\cssId{sb119}{\tan(x) := \frac{\sin(x)}{\cos(x)}}$$

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
