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← Everything for Calculus I
Calculus IApplications

Linear approximation, small changes made simpler.

Zoom in on a tangent line, estimate nearby values, and see how small measurement errors affect an answer. Then practise every class example, one step at a time.

A nearby straight line · Step 1 of 5

A smooth curve looks straight up close

Burgundy: square root · Dashed blue: tangent

361.52.25x
A tangent line touches a curve at a chosen input a{a} and has the same slope there. If the derivative exists, that line can estimate the curve’s height nearby. This is called linear approximation.
Try it: Zoom in around a=4{a} = {4} on f(x)=x{f(x)} = {\sqrt{x}}. Watch the gap between the curve and its tangent shrink. They look alike close up, but they are different functions.
ZoomWide
Zoom outZoom in
L(x)=f(a)+f′(a)(x−a){\displaystyle L(x)} = {\displaystyle f(a)} + {\displaystyle f'(a)(x-a)}

The starting height is f(a){f(a)}. Add the slope times the horizontal change.

f(4)=2,f′(4)=14{\displaystyle f(4)} = {\displaystyle 2,\qquad f'(4)} = {\displaystyle \frac14}
L(x)=2+14(x−4){\displaystyle L(x)} = {\displaystyle 2} + {\displaystyle \frac14(x-4)}

The line and curve agree exactly at the starting point. Away from it, use an approximation sign: f(x)≈L(x){f(x)} \approx { L(x)}.

Check yourself
What must the tangent and curve share at the starting point?

Need help with differentials or error estimates?

Bring your practice questions. We work through them together until every type feels routine.

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