Math Problem Statement

You said: \frac{1}{8}e^{2x}+e^x+x

calcule x=2 - x=1

Solution

We need to evaluate the function

f(x)=18e2x+ex+xf(x) = \frac{1}{8}e^{2x} + e^x + x

at x=2x = 2 and x=1x = 1, then find the difference f(2)f(1)f(2) - f(1).

Step 1: Compute f(2)f(2)

f(2)=18e4+e2+2f(2) = \frac{1}{8} e^{4} + e^2 + 2

Step 2: Compute f(1)f(1)

f(1)=18e2+e+1f(1) = \frac{1}{8} e^{2} + e + 1

Step 3: Compute the difference

f(2)f(1)=(18e4+e2+2)(18e2+e+1)f(2) - f(1) = \left(\frac{1}{8} e^{4} + e^2 + 2\right) - \left(\frac{1}{8} e^{2} + e + 1\right)

=18e4+e2+218e2e1= \frac{1}{8} e^4 + e^2 + 2 - \frac{1}{8} e^2 - e - 1

=18e4+(e218e2)+(21)e= \frac{1}{8} e^4 + \left(e^2 - \frac{1}{8} e^2\right) + (2 - 1) - e

=18e4+78e2+1e= \frac{1}{8} e^4 + \frac{7}{8} e^2 + 1 - e

Now, let's calculate the numerical value. math

Define e (Euler's number)

e = math.e

Compute values

f_2 = (1/8) * e4 + e2 + 2 f_1 = (1/8) * e**2 + e + 1

Compute the difference

difference = f_2 - f_1 differenceThe computed value of f(2)f(1)f(2) - f(1) is approximately 11.572.

Would you like a step-by-step breakdown of the calculations?

Here are five related questions you might find interesting:

  1. What is the derivative of f(x)=18e2x+ex+xf(x) = \frac{1}{8}e^{2x} + e^x + x?
  2. How would the difference change if we evaluated x=3x = 3 and x=2x = 2 instead?
  3. What is the second derivative of f(x)f(x), and what does it tell us?
  4. How does the function behave for large values of xx?
  5. Can we approximate f(x)f(x) using a Taylor series near x=0x = 0?

Tip: Exponential functions grow rapidly, so small changes in xx can have a significant effect on function values!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Function Evaluation
Algebraic Simplification

Formulas

f(x) = \frac{1}{8}e^{2x} + e^x + x
f(a) - f(b) = \left(\frac{1}{8} e^{2a} + e^a + a\right) - \left(\frac{1}{8} e^{2b} + e^b + b\right)

Theorems

Properties of Exponential Functions
Basic Function Operations

Suitable Grade Level

Grades 10-12