The number of tangents y=e^|x|   at  (0, 1) 

The function given is \( y = e^{|x|} \), which is piecewise defined as:

\[
y = 
\begin{cases} 
e^x & \text{if } x \geq 0, \\
e^{-x} & \text{if } x < 0.
\end{cases}
\]

The point of interest is \( (0, 1) \), where \( x = 0 \). To find the number of tangents at this point, we must consider the derivatives from both sides of \( x = 0 \).

### Step 1: Compute the derivatives on both sides of \( x = 0 \)
- For \( x \geq 0 \), the function is \( y = e^x \).
    - The derivative is \( \frac{dy}{dx} = e^x \), so at \( x = 0 \), the slope is \( e^0 = 1 \).
  
- For \( x < 0 \), the function is \( y = e^{-x} \).
    - The derivative is \( \frac{dy}{dx} = -e^{-x} \), so at \( x = 0 \), the slope is \( -e^0 = -1 \).

### Step 2: Analyze the behavior at \( x = 0 \)
At \( x = 0 \), the function \( y = e^{|x|} \) is continuous and differentiable from both sides, but the slopes are different:
- The right-hand derivative at \( x = 0 \) is \( 1 \),
- The left-hand derivative at \( x = 0 \) is \( -1 \).

Since the left-hand and right-hand derivatives are not equal, the function has two different tangents at \( (0, 1) \).

### Conclusion:
The number of tangents to the curve \( y = e^{|x|} \) at \( (0, 1) \) is **2**.

Would you like more details or have any further questions? Here are some related questions to explore:

1. How do left-hand and right-hand derivatives affect differentiability at a point?
2. Can you generalize this result for other functions with absolute values?
3. What is the geometric significance of having two tangents at a point?
4. How do tangents behave for a smooth curve without sharp points?
5. How does the piecewise nature of a function influence its derivative?

**Tip:** When dealing with piecewise functions, always check both the left-hand and right-hand derivatives at critical points to understand the behavior of the function fully.

The number of tangents to the curve y = e^|x| at the point (0, 1).

Explore the problem of finding the number of tangents to the curve y = e^|x| at the point (0, 1). This solution covers piecewise functions, left-hand and right-hand derivatives, and how differentiability influences tangent lines. Suitable for high school calculus students in Grades 11-12.

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