Build a tangent model, quantify its error in a worked case and distinguish small input changes from globally linear behavior.
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Try
Without multiplying 3.1 by itself, estimate its square using the known value 3²=9 and slope 2·3=6. An input increase of 0.1 suggests an output increase of 6·0.1=0.6, giving 9.6. Now calculate the exact square, 9.61. Where did the missing 0.01 come from?
Predict
Predict whether the tangent estimate to x² will be too high or too low for an input just below the base as well as just above it. Then predict how the error changes if the input step is halved.
Experiment
Write x=a+h. Expanding gives (a+h)²=a²+2ah+h². The first two terms form the tangent prediction; the last term is its exact remainder.
Base a
Increment h
Tangent prediction a²+2ah
Exact value
Exact minus predicted
3
0.1
9.6
9.61
0.01
3
−0.1
8.4
8.41
0.01
3
0.05
9.3
9.3025
0.0025
3
1
15
16
1
Draw y=x² and the line y=9+6(x−3) near x=3. Label the common point, slope and vertical prediction error at x=3.1. Use the exact remainder to explain why the tangent lies below this particular curve on both sides, and why halving the step quarters this particular error.
Understand
For a differentiable function, the linearization at a is L(x)=f(a)+f′(a)(x−a). It passes through (a,f(a)) and has the same slope as the function there. If x is close to a, L(x) provides a first-order prediction of f(x). The term “first-order” means the prediction includes the part proportional to the small input change.
The derivative definition says the remainder divided by the input increment tends to zero as that increment tends to zero. It does not, by itself, specify a numerical error bound at a particular nonzero increment. For x² we have additional algebra proving the remainder h². Do not assume all functions have that exact error or that every tangent underestimates.
For √x at a=9, the known value is 3 and derivative is 1/(2√9)=1/6. Thus L(x)=3+(x−9)/6. At x=9.6 the prediction is 3.1. Squaring 3.1 gives 9.61, so the estimate is not exact. The base 9 is chosen because both its square root and slope are manageable; choosing a nearby convenient base is part of the method.
Units remain meaningful: if f measures metres and x seconds, f′(a)(x−a) multiplies metres per second by seconds, giving an output correction in metres. Adding that correction to f(a) is dimensionally consistent.
Apply
The practice bank asks for a tangent prediction, an exact error, an irrational-value estimate and a justified error radius. Distinguish each request. For the quadratic error-radius task, show both that the proposed radius works and that any larger radius allows an error beyond the requirement. For a general unknown function, state what additional information would be needed before promising a finite-step error tolerance.
Continue
This completes ten lessons on calculus foundations, not a full Calculus I or AP course. Next steps include broader derivative rules, theorems about change, optimization and integral accumulation. Preserve the distinction between finishing these exercises, independently explaining their reasoning and demonstrating the wider course outcomes.
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Practice and reasoning
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For f(x)=x², use the tangent at a=3 to predict f(3.1). Enter the tangent prediction, not the exact square.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For f(x)=x², the tangent at a=4 predicts f(4.2). What is exact value minus tangent prediction?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Use the tangent to √x at x=9 to estimate √9.6. Give the tangent equation and distinguish an estimate from an exact value.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
You know only f(2)=5 and f′(2)=−3. Which claim is justified?
For f(x)=x², choose the largest positive δ ensuring the tangent prediction at any base a has absolute error at most 0.01 whenever |x−a|≤δ. Justify that the result does not depend on a.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
