My teacher showed our class how to find the graph of a derivative using tangent lines. I can not figure this out.
I would really appreciate it if someone could help me or explain it to me.
Thank you very much for your help.
The slope of a straignt tangent line at any point of a y(x) curve is the derivative of the y(x) function at that point.
To get the derivative this way, you would have to draw a series of tangent lines at diffent points of the curve, measure the slope of each line, and replot the derivative function (slope vs. x) using those measured points.
Thank you very much, drwls.
I actually understand it now!
(You saved me in Calculus!) :-)
Of course, I'd be happy to explain how to find the graph of a derivative using tangent lines!
To begin, let's start with a brief overview of derivatives. The derivative of a function measures its rate of change at each point. It gives us information about how the function is behaving.
One way to find the graph of a derivative is to use tangent lines. Tangent lines provide an approximation of the derivative of a function at a specific point. By drawing multiple tangent lines at different points on the original function's graph, we can get an idea of how the derivative behaves across the entire function.
Here's a step-by-step guide on how to find the graph of a derivative using tangent lines:
1. Select a function for which you want to find the derivative graph.
2. Choose a point on the function's graph where you want to determine the derivative. This point will be the center of the tangent line.
3. Identify the slope of the tangent line at that point. To do this, choose two points (on either side of the central point) that are very close together. Then, calculate the slope of the line passing through these two points. The closer the points are, the more accurate the derivative estimation will be.
4. Repeat steps 2 and 3 for various points along the original function's graph. Be sure to choose points that are representative of the entire function. The more tangent lines you draw, the more accurate your derivative graph will be.
5. Once you have determined the slopes for several tangent lines at different points on the function's graph, plot these slopes as points on a new graph. The x-coordinate of each point will correspond to the x-coordinate of the original function's graph, while the y-coordinate will represent the slope value of the corresponding tangent line.
6. Connect the plotted points on the new graph to form a smooth curve. This curve will represent the derived function graph and can be interpreted as the rate of change of the original function.
By following these steps, you can use tangent lines to estimate and visualize the graph of the derivative of a function.