AP Calculus AB Linearization and L’Hopital’s Rule Lesson (Free PDF)
This linearization lesson for AP Calculus AB covers linearization and l'hopital's rule (Common Core AP CALC AB 4.5, AP CALC AB 4.6, AP CALC AB 4.7), with a free PDF to print.
A local linear approximation uses the tangent line L(x) = f(a) + f′(a)(x – a) to estimate f near x = a. It overestimates when the graph is concave down and underestimates when it is concave up. L’Hôpital’s Rule evaluates limits of the form 0/0 or ∞/∞ by differentiating the numerator and denominator separately.
AP Calculus AB · Common Core State Standards: AP CALC AB 4.5, AP CALC AB 4.6, AP CALC AB 4.7
Key facts in this linearization lesson
- Linearization: L(x) = f(a) + f′(a)(x – a), the tangent line at x = a.
- Concave up (f″ positive): the tangent line is below the curve, so the estimate is too low.
- Concave down (f″ negative): the tangent line is above the curve, so the estimate is too high.
- L’Hôpital: for 0/0 or ∞/∞, lim f/g = lim f′/g′ (not the quotient rule).
What is a local linear approximation?
Near a point \(x=a\), a differentiable curve is almost a straight line: its tangent line. The tangent line gives a simple formula that approximates the function for inputs close to \(a\).
\(L(x)=f(a)+f^{\prime}(a)(x-a)\), and \(f(x)\approx L(x)\) for \(x\) near \(a\). Choose \(a\) so that \(f(a)\) and \(f^{\prime}(a)\) are easy to compute.
Example. Estimate \(\sqrt{4.5}\). Take \(f(x)=\sqrt{x}\), \(a=4\), \(f(4)=2\) and \(f^{\prime}(4)=\frac{1}{4}\). Then \(L(x)=2+\frac{1}{4}(x-4)\) and \(L(4.5)=2.125\). The calculator value is \(2.1213\), so the estimate is slightly high.
When is the estimate too high or too low?
Check the concavity. If \(f^{\prime\prime} > 0\) (concave up) near \(a\), the tangent line lies below the curve, so \(L(x)\) is an underestimate. If \(f^{\prime\prime} < 0\) (concave down), the tangent line lies above, so it is an overestimate. Square root is concave down, which explains the example.
The approximation gets better as \(x\) gets closer to \(a\). For \(\sqrt{4.5}\) the error is about \(0.004\), but using the same tangent line to estimate \(\sqrt{9}\) gives \(2+\frac{5}{4}=3.25\), which is far from the true value \(3\). Always stay close to the point of tangency, and choose a point where the function value is a nice number such as a perfect square or cube.
What is L’Hôpital’s Rule?
If \(\lim_{x\to a}\frac{f(x)}{g(x)}\) has the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}\) provided the second limit exists (or is infinite).
Example. \(\lim_{x\to 0}\frac{e^{x}-1}{x}\) is \(\frac{0}{0}\). Differentiate top and bottom: \(\lim_{x\to 0}\frac{e^{x}}{1}=1\). Another: \(\lim_{x\to\infty}\frac{x}{e^{x}}\) is \(\frac{\infty}{\infty}\), and \(\lim\frac{1}{e^{x}}=0\).
Differentiate the numerator and denominator separately; do not use the quotient rule. If the new limit is still indeterminate, apply the rule again. Never use the rule when the form is not indeterminate: \(\lim_{x\to 0}\frac{x+2}{x-1}=-2\) by substitution.
Linearization lesson: questions and answers
What is a linear approximation?
It is the tangent line used to estimate a function near the point of tangency. The formula is L(x) = f(a) + f′(a)(x – a).
How do you tell if a linear approximation is an overestimate?
Check the concavity. If the function is concave down near a, the tangent line lies above the graph, so the approximation is an overestimate. Concave up gives an underestimate.
When can you use L’Hôpital’s Rule?
Only when substituting gives the indeterminate form 0/0 or infinity over infinity. If the form is anything else, evaluate the limit another way.
Is L’Hôpital’s Rule the same as the quotient rule?
No. L’Hôpital differentiates the numerator and denominator separately and then takes a limit. The quotient rule differentiates the whole fraction.
What if the limit is still 0/0 after one application?
Apply the rule again. Keep differentiating the numerator and denominator until the form is no longer indeterminate.
How to use this linearization lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the linearization lesson as a PDF with the button below and keep it with your notes.
Download the linearization lesson as a PDF with the button below and keep it with your notes.
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