MathematicsChapter 2: Limit, Continuity and Derivative
Tangent, Normal, and Rate of Change
At x=a, derivative gives tangent slope; normal slope is -1/m when m is nonzero.
Learning objectives
- State the rule for tangent, normal, and rate of change.
- Solve a tangent, normal, and rate of change problem step by step.
- Check the result, restrictions, notation, and units.
Definition
At x=a, derivative gives tangent slope; normal slope is -1m when m is nonzero.
Formula
y=x2 at x
Where / symbols
- Tangent, normal, and rate of change
- At x=a, derivative gives tangent slope; normal slope is -1/m when m is nonzero.
CONCEPT AND RULE
At x=a, derivative gives tangent slope; normal slope is -1m when m is nonzero.
METHOD
For tangent, normal, and rate of change, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
For y=x^2 at x=2, tangent slope is 4 and tangent is y-4=4(x-2).
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve tangent, normal, and rate of change
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- For y=x^2 at x=2, tangent slope is 4 and tangent is y-4=4(x-2).
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Tangent, normal, and rate of change
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Tangent, Normal, and Rate of Change
Sticky Notes · Formula and Worked Revision
Core Idea
- At x=a, derivative gives tangent slope; normal slope is -1m when m is nonzero.
Rule / Formula
- At x=a, derivative gives tangent slope; normal slope is -1m when m is nonzero.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For y=x^2 at x=2, tangent slope is 4 and tangent is y-4=4(x-2).
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- At x=a, derivative gives tangent slope; normal slope is -1m when m is nonzero.
- Worked pattern: For y=x^2 at x=2, tangent slope is 4 and tangent is y-4=4(x-2).
- Final check: domain, sign, notation, units, and reasonableness.
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