https://file.aiquickdraw.com/mathbot/file/1727893341995lcqeu3n1.jpg

To address the question, let's break down the problem mathematically.

### Step 1: Find the equation of the tangent line at \(x = 1.5\)
The function given is \( f(x) = e^x \). The derivative of \(f(x)\) gives the slope of the tangent line:

\[
f'(x) = \frac{d}{dx}(e^x) = e^x
\]

At \(x = 1.5\), the slope of the tangent line is:

\[
f'(1.5) = e^{1.5} \approx 4.4817
\]

The point on the curve where \(x = 1.5\) is:

\[
f(1.5) = e^{1.5} \approx 4.4817
\]

Thus, the point of tangency is \((1.5, 4.4817)\), and the slope of the tangent line is approximately 4.4817.

### Step 2: Equation of the tangent line at \(x = 1.5\)
Using the point-slope form of the equation of a line \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is \((1.5, 4.4817)\) and \(m = 4.4817\):

\[
y - 4.4817 = 4.4817(x - 1.5)
\]

Simplifying:

\[
y = 4.4817(x - 1.5) + 4.4817
\]
\[
y = 4.4817x - 6.7226 + 4.4817
\]
\[
y = 4.4817x - 2.2409
\]

### Step 3: Check if the tangent line passes through \((0.5, 0)\)
Substitute \(x = 0.5\) into the equation of the tangent line:

\[
y = 4.4817(0.5) - 2.2409 = 2.24085 - 2.2409 \approx 0
\]

Thus, the tangent line at \(x = 1.5\) **does pass through the point \((0.5, 0)\)**. The student is correct.

### Step 4: Tangent line at \(x = 2.5\)
First, we find the slope of the tangent line at \(x = 2.5\):

\[
f'(2.5) = e^{2.5} \approx 12.1825
\]

The point on the curve where \(x = 2.5\) is:

\[
f(2.5) = e^{2.5} \approx 12.1825
\]

Thus, the point of tangency is \((2.5, 12.1825)\), and the slope is approximately 12.1825. Using the point-slope form of the equation:

\[
y - 12.1825 = 12.1825(x - 2.5)
\]

Simplifying:

\[
y = 12.1825(x - 2.5) + 12.1825
\]
\[
y = 12.1825x - 30.45625 + 12.1825
\]
\[
y = 12.1825x - 18.27375
\]

### Step 5: Check if the tangent line passes through \((1.5, 0)\)
Substitute \(x = 1.5\) into the equation of the tangent line:

\[
y = 12.1825(1.5) - 18.27375 = 18.27375 - 18.27375 = 0
\]

Thus, the tangent line at \(x = 2.5\) **does pass through the point \((1.5, 0)\)**.

### Conclusion:
- Yes, the student's claim that the tangent line at \(x = 1.5\) passes through \((0.5, 0)\) is correct.
- The tangent line at \(x = 2.5\) also passes through the point \((1.5, 0)\).

Would you like more details or have any questions? 

Here are some related questions:
1. What are the general properties of the exponential function \(e^x\)?
2. How do derivatives determine the slope of a tangent line?
3. Why does the tangent line at a point only touch the curve at that specific point?
4. Can two different tangent lines pass through the same point on a graph?
5. What is the geometric significance of a line tangent to a curve?

**Tip:** When solving problems involving tangent lines, the point-slope form is a quick and effective way to find the equation of the line once the slope and a point are known.

A student claims that the line tangent to the graph of f(x) = e^x at x = 1.5 passes through the point (0.5,0). Is she correct? Will the line tangent at x = 2.5 pass through (1.5,0)? Explain.

Explore whether the tangent line to the exponential function f(x) = e^x at x = 1.5 passes through the point (0.5,0). Learn how to calculate the equation of the tangent line and verify the claim. This problem involves calculus concepts such as derivatives and exponential functions, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation