Linear approximation uses the tangent line to estimate the value of a function near a known input. This cheat sheet helps students quickly connect tangent lines, derivatives, differentials, and small changes in function values. It is useful for estimating values without a calculator and for understanding how derivatives describe local behavior.
These ideas also prepare students for related rates, optimization, and error analysis.
Key Facts
- The linear approximation of near is .
- For a small change , the change in the function is approximated by .
- The differential of is , and the differential of is .
- Near , the function value can be estimated by .
- The tangent line approximation is most accurate when is close to and is differentiable at .
- Absolute error can be estimated by , where .
- Relative error is , and percent error is .
- If is large in magnitude near , the tangent-line approximation may become less accurate more quickly.
Vocabulary
- Linear approximation
- An estimate of a function near a point using the tangent line formula .
- Tangent line
- A line that touches a curve at a point and has slope equal to the derivative at that point.
- Differential
- A small estimated change in a dependent variable, written as .
- Increment
- A change in an input or output value, often written as or .
- Error estimate
- A measure of how far an approximation is from the actual value, often written as .
- Local linearity
- The idea that a differentiable function looks nearly like a straight line when viewed very close to a point.
Common Mistakes to Avoid
- Using instead of in is wrong because the slope of the tangent line is fixed at the base point .
- Forgetting to choose a nearby convenient value for leads to poor estimates because linear approximation works best when is close to .
- Confusing with is incorrect because is the actual change while is the tangent-line estimate.
- Dropping the sign of can reverse the direction of the estimate, since depends on whether the input increases or decreases.
- Using linear approximation far from the tangent point gives unreliable results because curve bending makes the tangent line less representative.
Practice Questions
- 1 Use linear approximation to estimate by choosing and .
- 2 For , use and to estimate with .
- 3 A sphere has radius cm with a possible measurement error of cm. Use to estimate the possible error in volume.
- 4 Explain why linear approximation is usually more accurate near the point of tangency than far away from it.
Understanding Linear Approximation & Differentials
A curve can look almost straight when viewed over a tiny interval. This local straightness is the main reason a first degree estimate works. The derivative gives the slope at one chosen point, so it tells how rapidly the output is changing at that instant.
A positive slope means nearby output values tend to rise as the input rises. A negative slope means they tend to fall. A slope of zero does not mean the function stays flat for long.
It only means the graph is level at that particular point. The curve may turn upward, downward, or change direction soon after.
Differentials give a practical language for small measurement changes. Think of dx as a planned or measured change in an input. The resulting dy is the change predicted from the local slope.
In science, inputs often have unavoidable uncertainty. A ruler may be accurate only to the nearest millimeter. A thermometer may have a small reading error.
If a formula uses those measurements, the differential estimates how much the final result could change because of the input uncertainty. This is useful for quantities such as the area of a circular object, the volume of a container, or the speed calculated from distance and time. The estimate is strongest when the measurement error is small compared with the original measurement.
The gap between a prediction and the true value comes from curvature. A straight line has no curvature, so its local estimate stays exact everywhere. Most functions bend, which causes the tangent line to drift away from the graph as the input moves farther from the chosen point.
The second derivative describes this bending. When a graph is concave up near the point, its tangent line usually lies below the graph nearby, so the estimate tends to be too small. When a graph is concave down, the estimate tends to be too large.
This pattern helps students check whether an answer is sensible before finding an exact value. Large curvature means that even a modest input change can create noticeable error.
Careful setup matters more than fast arithmetic. Choose a known input that is close to the value you need and that makes the function easy to evaluate. Keep track of whether the input change is positive or negative.
Use units throughout the work. If the input is measured in meters, the predicted output change must have the output unit, such as square meters for area or cubic meters for volume. Distinguish the actual change from the predicted change.
They are related but not identical. For percent error, compare the size of the error with the size of the actual quantity, then multiply by one hundred. This comparison is especially important when the actual quantity is very small, since a small absolute error can become a large percent error.