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.