Math Problem Statement
If
w ′(t)
is the rate of growth of a child in pounds per year, what does
11
8
w ′(t) dt
represent?
The child's initial weight at birth.The child's weight at age 11. The change in the child's weight (in pounds) between the ages of 8 and 11.The change in the child's age (in years) between the ages of 8 and 11.The child's weight at age 8.
Solution
The expression
represents the change in the child's weight (in pounds) between the ages of 8 and 11.
Explanation:
- The integral of a rate of change, , over a specific interval gives the total change in the quantity over that interval.
- Here, is the rate of growth in pounds per year, and the integral from to sums up all the changes in the child's weight over this period.
- Therefore, the result of this integral gives the total change in the child's weight between ages 8 and 11.
Correct Answer:
The change in the child's weight (in pounds) between the ages of 8 and 11.
Would you like further details or have any questions? Here are 5 related questions for you to explore:
- What is the physical interpretation of a definite integral in terms of real-world problems?
- How would the interpretation change if were a negative value over part of the interval?
- What if the integral limits were reversed, such as ?
- How would you calculate if you were given and ?
- Can you explain the connection between the Fundamental Theorem of Calculus and this problem?
Tip: Always interpret the units of the rate and the limits of integration to understand the physical meaning of an integral.
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Math Problem Analysis
Mathematical Concepts
Definite Integration
Rate of Change
Physical Interpretation of Integrals
Formulas
∫ₐᵇ f'(x) dx = f(b) - f(a)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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