Math Problem Statement

Consider the function f(x) = 7^x. Given a table with x-values that get closer to π ≈ 3.14159265359 and their corresponding f(x) values. Use the table to approximate 7^π correct to four decimal places.

Solution

The table provides values of the function f(x)=7xf(x) = 7^x for xx values approaching π\pi. We need to approximate 7π7^\pi to four decimal places, using the closest approximation of π3.14159265359\pi \approx 3.14159265359.

Let's look at the table:

  • The value closest to π\pi is x=3.14159265x = 3.14159265.
  • The corresponding value of f(x)=7xf(x) = 7^x is 451.8079451.8079.

Thus, the approximation of 7π7^\pi, correct to four decimal places, is:

7π451.80797^\pi \approx 451.8079

Would you like more details on this problem, or do you have any questions?

Here are 5 related questions to expand on the topic:

  1. How is the value of π\pi determined accurately in mathematics?
  2. Why does 7x7^x grow so quickly compared to linear functions?
  3. How can we calculate powers of irrational numbers like π\pi more precisely?
  4. What methods are available for approximating the value of f(x)f(x) when exact computation isn't possible?
  5. What are some real-world applications where calculating powers involving π\pi is important?

Tip: When approximating functions using tables, try to find values that are very close to the target to minimize error.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Approximations
Irrational Numbers

Formulas

f(x) = 7^x

Theorems

Exponential Growth

Suitable Grade Level

Grades 9-12