The Fundamental Theorem and accumulated change
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| accumulation function/əˌkjuːmjʊˈleɪʃn ˈfʌŋkʃn/ | 累积函数 | lěi jī hán shù |
A flow meter reports a changing rate. How does the accumulated volume recover that same rate when differentiated?
- A flow meter reports a changing rate. How does the accumulated volume recover that same rate when differentiated?
- This lesson studies accumulation function 累积函数: A function whose value is the signed integral from a fixed starting point to a variable endpoint.
Choose the mathematical structure
- For continuous f on an interval, A(x)=integral from a to x of f(t) dt has A′(x)=f(x). If F′=f, the definite integral from a to b is F(b)−F(a). These two directions connect accumulation and differentiation. Dummy variable t is integrated; x is the moving endpoint. For A(g(x)), apply the chain rule: its derivative is f(g(x))g′(x).
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines accumulation function?
A function whose value is the signed integral from a fixed starting point to a variable endpoint.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Let A(x)=integral from 1 to x of (3t²−2) dt. An antiderivative is t³−2t, so A(x)=x³−2x+1, A(1)=0, A(2)=5 and A′(2)=10. Thus accumulated change 5 differs from endpoint rate 10. For B(x)=integral from 1 to x² of (3t²−2) dt, B′(x)=(3x⁴−2)2x; at x=1 this is 2. Independently, if F′=3x²−2 and F(1)=4, then F=x³−2x+5. Its change F(2)−F(1)=5 agrees with A(2), although its initial value is not zero.
The Fundamental Theorem and accumulated change
For continuous f on an interval, A(x)=integral from a to x of f(t) dt has A′(x)=f(x)
Check the hypothesis, endpoints and coefficients behind each integral calculation.
Find A(2) for the worked accumulation function.
A(2)=8−4+1=5.
Test a tempting shortcut
- Subtract the lower antiderivative value even when the lower limit is not zero. A definite integral is signed change, not automatically total positive area. A′(x) is the integrand at the endpoint, not its antiderivative. Continuity on the relevant interval is a sufficient hypothesis; do not integrate across a pole such as 1/t at zero. A moving endpoint x² also contributes its chain-rule factor.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of an accumulation function is always its accumulated total. This claim is false. Explain which definition or assumption it violates.
Find A′(2).
A′(2)=3×4−2=10.
The derivative of an accumulation function is always its accumulated total.
Subtract the lower antiderivative value even when the lower limit is not zero. A definite integral is signed change, not automatically total positive area. A′(x) is the integrand at the endpoint, not its antiderivative. Continuity on the relevant interval is a sufficient hypothesis; do not integrate across a pole such as 1/t at zero. A moving endpoint x² also contributes its chain-rule factor.
Interpret a new situation
- Use rate units multiplied by input units for accumulated change. Check A(a)=0 and differentiate the final expression to recover f. Initial values determine constants in a state function; definite changes do not depend on an arbitrary antiderivative constant. This lesson gives the theorem and elementary variable-bound examples; improper integrals are outside this treatment.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find B′(1) for the worked moving endpoint.
B′(1)=(3−2)×2=2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · H. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A function whose value is the signed integral from a fixed starting point to a variable endpoint. Choose the relationship, show the method, check its assumptions and interpret the result.