help needed!!!
how would you go about drawing the graph of a functions derivative, if the original function is unknown. IE only the graph if F(x) is given but not the equation.
thanks
You could construct the tangent at different points, and if the graph is free of kinks and vertical asymptotes, you can join the different points to get a graph of the derivative.
A kink will imply a discontinuity in the derivative, and a vertical asymptote will mean the same thing in the derivative.
I hope this just about covers most situations.
Addendum:
"You could construct the tangent at different points..."
should read:
"You could construct and plot the magnitude of the tangents at different points..."
If you are given the graph of the function f(x) and want to draw the graph of its derivative, you can use the following steps:
1. Identify key points on the graph of f(x). Look for any points where the slope of the tangent line appears to change significantly. These points will correspond to the x-values of critical points, such as local maxima, local minima, and points of inflection.
2. Determine the slopes of the tangent lines at these key points. Estimate the slopes by using the rise-over-run method. Select two nearby points on the curve and calculate the change in y divided by the change in x. Repeat this process for different pairs of nearby points to get an idea of the overall trend of the slope.
3. Plot the points on the graph of the derivative. The x-values of the critical points should correspond to the x-values of the key points on the graph of f(x). The y-values will be the slopes of the tangent lines at those points.
4. Connect the plotted points with smooth curves. Use the shape of the tangents to guide the shape of the derivative curve. If the tangent is upward-sloping, the derivative should be positive. If the tangent is downward-sloping, the derivative should be negative. Also, consider the concavity of f(x) at each point to determine whether the derivative is increasing or decreasing.
Keep in mind that these steps are an estimation, and the accuracy of the resulting graph will depend on the quality of the original graph of f(x). Additionally, the derivative graph will not provide all the information about the original function, such as exact formulas or specific behavior between critical points.