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The tangent to the curve y=e 2x at the point (0,1) meets x-axis at?

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Final answer:

The tangent to the curve y=e^{2x} at the point (0,1) meets the x-axis at the point (-1/2, 0). To determine this, the derivative of the curve was used to find the slope of the tangent, which was then used to find the tangent line equation. Setting the y-coordinate to zero gave the x-intercept of the tangent line.

Step-by-step explanation:

The student is asking about the point where the tangent to the curve y=e^{2x} at the point (0,1) intersects the x-axis. To find the point of intersection, we first need to determine the slope of the tangent to the curve at that point. The slope of a curve at a specific point is given by its derivative at that point.

We can calculate the derivative of y=e^{2x} as follows:

y' = d/dx (e^{2x}) = 2e^{2x}.

At the point (0,1), the slope of the tangent is therefore y'(0) = 2e^{2(0)} = 2.

The equation of the tangent line can be written in the slope-intercept form as:

y = mx + c,

where m is the slope and c is the y-intercept. Since the line is tangent to the curve at (0,1), the point (0,1) lies on the line, which gives us the equation:

y = 2x + 1.

To find the x-coordinate of the point where this line meets the x-axis, we need to set y to 0 (since any point on the x-axis has a y-coordinate of 0) and solve for x:

0 = 2x + 1,

x = -1/2.

Therefore, the tangent meets the x-axis at the point (-1/2, 0).

User Brutal
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