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Answer

y=−x+1y = -x + 1

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Okay, Victor. Let's tackle these questions in English.

Here's the solution for Question 1:

  1. Find the y-coordinate of the point on the curve where x=1x=1.

    Substitute x=1x=1 into the equation of the curve y=x2−3x+2y = x^2 - 3x + 2:

    y=(1)2−3(1)+2y = (1)^2 - 3(1) + 2 y=1−3+2y = 1 - 3 + 2 y=0y = 0

    So, the point of tangency is (1,0)(1, 0).

  2. Find the derivative of the curve to get the slope function.

    Differentiate y=x2−3x+2y = x^2 - 3x + 2 with respect to xx:

    dydx=ddx(x2)−ddx(3x)+ddx(2)\frac{dy}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(3x) + \frac{d}{dx}(2) dydx=2x−3\frac{dy}{dx} = 2x - 3
  3. Evaluate the derivative at x=1x=1 to find the slope of the tangent line.

    Substitute x=1x=1 into the derivative:

    m=2(1)−3m = 2(1) - 3 m=2−3m = 2 - 3 m=−1m = -1
  4. Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1), to find the equation of the tangent.

    Using the point (x1,y1)=(1,0)(x_1, y_1) = (1, 0) and the slope m=−1m = -1:

    y−0=−1(x−1)y - 0 = -1(x - 1) y=−x+1y = -x + 1

    The equation of the tangent to the curve at the point where x=1x=1 is y=−x+1\boxed{y = -x + 1}.

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