Student Resources
Estimates on a graph
For the function graphed below, we only know its derivative:
\(\displaystyle f'(x)=2e^{-(x-2)^2}-\frac{x}{10}\)
and the only known value is \(f(2)=1.5\).
Nobody has ever found another exact value of this function! The graph provided is only an approximation and should not be trusted for exact values.
Once you enter the correct value of \(f'(2)\), the tangent line will appear. You will then be able to move the blue point to determine corresponding changes \(dx\) and \(dy\) on the tangent line. You can also zoom in and out as needed. The circular arrows in the upper right reset the entire graph.
Click ⛶ to open in full screen
Find the value of \(f'(2)\) and enter it in the interactive graph above.
Once the tangent line appears, drag the blue dot to use the linearization to estimate \(f(1.5)\)
Based on the shape of the graph, are the slopes of tangent lines increasing or decreasing for \(1.5\leq x\leq2\)? Does this imply your approximation \(f(1.5)\) is an underestimate or overestimate?
Drag the blue dot to use the linearization to estimate the value of \(x\) for which \(f(x)=2\).
Based on the shape of the graph, are the slopes of tangent lines increasing or decreasing for \(2\leq x\leq3\)? Does this imply your approximation for the value of \(x\) for which \(f(x)=2\) is an underestimate or overestimate?