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

To determine the number of tangents to the curve \( y = e^{|x|} \) at the point \((0, 1)\), we need to follow these steps:

### Step 1: Analyze the Function
The function is \( y = e^{|x|} \), which can be split into two cases:

- For \( x \geq 0 \), \( y = e^x \).
- For \( x < 0 \), \( y = e^{-x} \).

### Step 2: Determine the Derivatives
The derivative \( \frac{dy}{dx} \) will give us the slope of the tangent to the curve:

- For \( x > 0 \), the derivative is \( \frac{dy}{dx} = e^x \).
- For \( x < 0 \), the derivative is \( \frac{dy}{dx} = -e^{-x} \).

### Step 3: Evaluate at the Point \((0, 1)\)
At \( x = 0 \), we need to check the slope of the tangents:

- For \( x = 0^+ \) (approaching from the right), the slope is \( \frac{dy}{dx} = e^0 = 1 \).
- For \( x = 0^- \) (approaching from the left), the slope is \( \frac{dy}{dx} = -e^0 = -1 \).

### Step 4: Conclusion
Since the slopes from the left and right of \( x = 0 \) are different (\( 1 \) and \( -1 \), respectively), there are two distinct tangents at the point \( (0, 1) \).

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

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. What is the geometric significance of having different slopes for tangents at a point?
2. How do we find the equation of the tangents at \( x = 0 \)?
3. How does the behavior of the function \( y = e^{|x|} \) differ from \( y = e^x \) alone?
4. Can a function have more than two tangents at a single point? If yes, under what conditions?
5. What is the graphical interpretation of the point \( (0, 1) \) on the curve \( y = e^{|x|} \)?

**Tip:** Understanding the behavior of derivatives on either side of a point is crucial in determining the number of tangents and their respective slopes.

Discover how many tangents exist to the curve y = e^|x| at the point (0, 1) through a detailed analysis of derivatives and absolute value function properties. Understand the distinct slopes and their significance in calculus. Suitable for college-level calculus students.

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