Tangent Line Zoom
Zoom in on a smooth curve and it starts to look like a straight line: its tangent line. That is why the tangent line gives good approximations near the point. See how big the error is, and use concavity to tell whether the approximation is too big or too small.
Drag across the small graph to move a.
Drag across the graph to move b.
Drag up or down on the graph to move k.
How straight does it look?
Right now
The whole graph
The box shows the part you are zoomed in on.
Try this
Picture it
Sketch the curve and the tangent line yourself first. Then compare.
Key ideas
- Tangent line at x = ay = f(a) + f′(a)(x − a)
- Approximation near af(b) ≈ f(a) + f′(a)(b − a)
- Concave upThe tangent line lies below the graph, so the approximation is an underestimate.
- Concave downThe tangent line lies above the graph, so the approximation is an overestimate.
- Justify with the whole intervalGive the sign of f″, or say f′ is increasing or decreasing, for every x between a and b.
Every problem is generated fresh, and every answer key is computed in code and checked with a computer algebra system. Scores stay in this browser only. Non-calculator. Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization. Over- and underestimates use concavity, which Topic 5.6 develops fully.