The general equation of a tangent line to a curve
Tangent line to function, f, through point, (a, b), where f ' (a) is the derivative of f at a, in point-slope form:
Please make sure to note that the slope position of point-slope form has been replaced with the derivative of f evaluated at a.
interactive visualizations
The following site provides an original function graph and you drag the slider, it provides the derivative value (slope of the tangent line) at that x-value and graphs the derivative function simultaneously:
Derivative Plotter
Here is a similar site on which you can choose whether the derivative function is graphed or not:
Derivative Graphs
On this site, you create your own function and it will calculate the actual value of the derivative at any requested x-value. You will have to be somewhat familiar with computer syntax (for example: use "^" to raise something to an exponent and "/" for division, and when in doubt, use parentheses), but it's still a neat tool.
Interactive Tangent Generator
Derivative Plotter
Here is a similar site on which you can choose whether the derivative function is graphed or not:
Derivative Graphs
On this site, you create your own function and it will calculate the actual value of the derivative at any requested x-value. You will have to be somewhat familiar with computer syntax (for example: use "^" to raise something to an exponent and "/" for division, and when in doubt, use parentheses), but it's still a neat tool.
Interactive Tangent Generator
Tangent line approximations
The tangent line value can be used to approximate the actual function value. However, as you move away from the point of tangency, the tangent line approximation decreases in accuracy.
![Picture](/uploads/2/6/0/9/26090300/1398296248.jpg)
Graph created using www.desmos.com.
Using the tangent line to approximate the function value may be preferred if the original function is complicated to evaluate, or when the original function is unknown. In the example above, we are given the original function in red and the line y = x is tangent to this function at x = 0. As the graph demonstrates, these two function values will be very similar close to the x-value of 0.
Let's say we need to know what the function value equals at x = -0.25. To evaluate this decimal x-value in the original function, we would need to raise to the fourth power, multiply by a negative, raise to the third power, multiply that by a -2, and add all of that together with the original value also (that's 9 multiplications and 3 additions.) It would take a lot longer than just using the tangent y-value, which equals the x-value, in this case. Therefore, a reasonable approximation of the original function height at x = -0.25 (using the tangent line equation instead) is y = -0.25. The actual function value, by the way, rounded to three places past the decimal, is y = -0.223. My approximated value had an error of 0.027.
It would not make sense, however, to use this tangent line to approximate the function value at x = -1, because the tangent line is very different from the function at this location (which is one unit away from the point of tangency.) Here, my tangent line approximation would be y = -1. My actual function value is y = 0. Generally, the tangent line approximation is much less reliable for x-values far away from the point of tangency.
Ready for practice?
This website provides 27 practice problems on writing the equation of the tangent line. Downloadable .pdf solutions for each problem are also available.
UIC Tangent Line Problem Practice
Here, you can find practice problems for a few more topics than just tangent line equations, but problem 2.3 and 2.4 are good ones to try for this topic.
wykaMath Applications of the Derivative Practice Problems
If you would like some additional explanation as well as examples and practice problems, check out this page:
17Calculus Tangent Lines
UIC Tangent Line Problem Practice
Here, you can find practice problems for a few more topics than just tangent line equations, but problem 2.3 and 2.4 are good ones to try for this topic.
wykaMath Applications of the Derivative Practice Problems
If you would like some additional explanation as well as examples and practice problems, check out this page:
17Calculus Tangent Lines