Calculus I · Unit 3A · lesson

Initial-Value Problems and Recovering a Function

Concept

Learning objectives

Use a derivative or rate together with one known value to recover a unique function.

Initial-Value Problems and Recovering a Function

Explanation

How one measured value selects one function

An antiderivative gives a family because a derivative cannot tell where the original graph sits vertically. An initial condition supplies exactly that missing information. Once a rate law has been integrated, substituting one known function value turns the arbitrary constant into a specific number and selects one member of the family. This is why a velocity law alone does not determine position, while velocity together with one known position does.

The same pattern appears far beyond motion. A growth rate plus one measured population, a cooling rate plus one measured temperature, or a marginal cost plus one known total cost can each determine a unique model. The practical workflow is always the same: integrate the rate, introduce the constant, apply the known value, and then verify both the derivative relation and the initial condition. Skipping either verification leaves room for a model that is algebraically tidy but physically wrong.

An antiderivative family becomes one specific function after an initial condition fixes the vertical shift.

Guided walkthrough

Recover a position function from velocity

A particle has velocity $v(t)=6t-4$ meters per second and position $s(2)=5$ meters. Find $s(t)$ .

Answer reveal

Worked solution

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Application

A temperature model from a measured cooling rate

Suppose a sensor records a temperature-change rate $T'(t)=-0.8e^{-0.2t}$ degrees Celsius per minute and $T(0)=24$ . Integration gives

T(t)=4e^{-0.2t}+C.

The initial value gives $24=4+C$ , so $T(t)=20+4e^{-0.2t}$ . The constant is not algebraic clutter; it identifies the ambient-temperature level approached by the model.

Interactive checku3a-ivp-01

If $F'(x)=6x^2-1$ and $F(0)=4$ , find $F(x)$ .

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Show hint

Integrate first, then substitute $x=0$ to determine $C$ .

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Exercise

If $y'=3x^2+2$ and $y(1)=7$ , find $y$ .

Exercise

A car has acceleration $a(t)=4t-2$ , velocity $v(0)=3$ , and position $s(0)=10$ . Find $v(t)$ and $s(t)$ .

Exercise

Explain why one initial value determines one antiderivative constant but a second-order problem generally requires two initial values.

After the explanation

Use the section idea

Reading lens

Connect every antiderivative to a derivative check, and every varying rate to a sum of rate-times-width contributions.

Mental model

Indefinite integration recovers a family of functions; definite accumulation combines signed local changes into one net change.

Decision

Ask whether the task wants a general antiderivative, an initial-condition solution, displacement, distance, or a numerical total from data.

Common trap

Omitting the arbitrary constant, confusing displacement with distance, or multiplying one changing rate by the entire interval.

Check yourself

Can you differentiate your antiderivative and interpret the sign and units of a rate-based total?

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