Math Problem Statement
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the given function at the given value. f left parenthesis x right parenthesis equals 2 x Superscript x; xequals2 Question content area bottom Part 1 The instantaneous rate of change for the function at xequals2 is
Solution
To find the instantaneous rate of change of the function at , we use the limit definition of the derivative, which is:
For at , this becomes:
Now, let's approximate this limit by using smaller and smaller values of . I'll calculate it for and to get progressively closer to the actual rate of change.
We know .
Now let's compute for different values of :
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For :
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For :
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For :
From these approximations, we see that the instantaneous rate of change approaches as becomes smaller. Hence, the instantaneous rate of change for at is approximately:
Would you like further details or have any questions?
Here are five related questions for you to consider:
- How does the derivative of exponential functions differ from polynomial functions?
- Can you explain why we use smaller and smaller values of for the approximation?
- What is the significance of the derivative in real-world applications?
- How would the derivative change if we used a different base, such as ?
- What does the instantaneous rate of change represent geometrically?
Tip: The derivative gives you the slope of the tangent line at any given point on the function graph.
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Math Problem Analysis
Mathematical Concepts
Calculus
Instantaneous Rate of Change
Limits
Exponential Functions
Formulas
f'(x) = lim(h → 0) [(f(x + h) - f(x)) / h]
f(x) = 2^x
Theorems
Limit Definition of Derivative
Suitable Grade Level
Grades 11-12, Early College
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