Determine whether each line is tangent to f at x = a, a secant line, or both a tangent line and a secant line.
Answer
This line is a secant line because it crosses the graph at two points. This line is not a tangent line because it crosses the graph at x = a rather than bouncing off.
This is a tangent line. The line bounces off the graph at x = a.
This is a tangent line. The line bounces off the graph at x = a without crossing it.
This is both a tangent line and a secant line. The line lies tangent to the graph at x = a, but then intersects the graph again in the first quadrant.
This is a tangent line. The line bounces off the graph at x = a. This is one of the rare occasions, though, where the line actually crosses the graph at the point of tangency.
Example 2
For the function f and point a, draw the tangent line to f at a:
Answer
Example 3
For the function f and point a, draw the tangent line to f at a:
Answer
Example 4
For the function f and point a, draw the tangent line to f at a:
Answer
Example 5
Determine if each line is tangent to f at a.
Answer
Yes. This line definitely "lays along" f. Since f is a line, the tangent line to f at a is the line f again.
This is a tangent line, just not at x = a. The line is tangent to graph at a point a little way to left of a.
This is a tangent line. The line just touches the graph at x = a and then bounces right off of it.
This is not a tangent line. The line passes right through the graph at x = a very non-tangentially.
This is not a tangent line. The line doesn't bounce off the graph at any point.
