Math Problem Statement
answer both questions
Solution
Let's solve both of the problems step by step:
Question 7:
Given:
The height of a projectile is represented by the function:
where is the height in meters, and is the time in seconds.
We need to find the average rate of change of height between and .
Formula for average rate of change:
The average rate of change of a function between and is given by: Here, and .
Step-by-step calculation:
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Find :
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Find :
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Use the average rate of change formula:
Thus, the average rate of change of the projectile's height between 0 and 2 seconds is 15.2 m/s.
Question 8:
Given:
We are asked to discuss how changing the value of in the difference quotient brings the slope of the secant closer to the slope of the tangent line. We are also asked if there is a limit to how small can be.
Explanation:
The difference quotient is the formula used to estimate the slope of the secant line between two points on a curve: As becomes smaller, the second point on the secant line moves closer to the first point, making the secant line better approximate the tangent line at .
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How changing affects the slope:
- As decreases, the two points (at and ) come closer together.
- This causes the secant line (which passes through both points) to better approximate the tangent line at .
- The smaller becomes, the closer the slope of the secant line becomes to the slope of the tangent line.
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Is there a limit to how small can be?
- As approaches 0, the secant line becomes indistinguishable from the tangent line, and the slope of the secant line approaches the derivative at , which is the slope of the tangent line.
- However, while conceptually can become arbitrarily small, in practice, there is a limit. If becomes too small (especially in numerical computations), round-off errors or precision issues can arise.
- Mathematically, approaching zero gives us the derivative, but itself is undefined in the difference quotient because it would involve division by zero.
In summary, the smaller gets, the closer the secant slope approximates the tangent slope, but there are practical limits due to numerical accuracy.
Would you like further clarification on any part of this? Here are 5 related questions you might want to explore:
- How is the difference quotient related to the derivative?
- Can you find the instantaneous rate of change of the projectile at seconds?
- What happens to the average rate of change if the interval is increased to seconds?
- Why do secant lines provide only an approximation of the tangent line’s slope?
- How does the second derivative relate to the curvature of the projectile's path?
Tip: When finding average rates of change, always remember that you're calculating the slope of the secant line between two points on the curve. For the instantaneous rate of change, use derivatives!
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Math Problem Analysis
Mathematical Concepts
Calculus
Rates of Change
Secant Line
Tangent Line
Formulas
Average rate of change = (h(t2) - h(t1)) / (t2 - t1)
Difference quotient = (f(x+h) - f(x)) / h
Theorems
Definition of Average Rate of Change
Limit Definition of the Derivative
Suitable Grade Level
Grades 11-12
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