Post #23

Tangent Lines

Tangent lines can be used for many purposes. Finding a tangent line requires only a little knowledge of calculus but a substantial amount of algebra.

Example:

Find the line tangent to the graph y=2x2+4x+6 at x=1.

1. Identify the equation and point of tangency. If not given a y value, plug the x value into the original equation.

y=2x2+4x+6 y=2(1)2

2. Differentiate you equation.

dy/dx=4x+4


3. Plug in x value then solve for dy/dx.


dy/dx=4(1)+4=8


Your dy/dx value is your slope from here on you can create your equation of the tangent line. There are three forms that your equation can be presented in: point-slope, slope-intercept, or standard form. For these purposes, I am using point-slope.


(y-12)=8(x-1) is your final answer.

Graph Interpretation

Every AP exam will include at least one graph on it's short answer section that requires the test-taker to interpret certain requirements from it in order to receive credit for that question.

Interpreting a graph is very easy but in order to do it properly one must understand certain properties.

1. If the original graph is increasing, the slope is positive.
2. If the original graph is decreasing, the slope is negative.
3. An interval with a positive slope on the first derivative means that there is a downward concavity on that interval in the second derivative.
4. An interval with a negative slope on the first derivative means that there is an upward concavity on that interval in the second derivative.
5. Upward concavity (bowl-shaped) is positive.
6. Downward concavity (umbrella-shaped) is negative.
7. There is a horizontal tangent where the slope=0 on the original graph.
8. Wherever there is a horizontal slope there is either a maximum or a minimum value.
9. A x intercept on the first derivative is either a maximum of a minimum on the original graph.