Finding the of the to a is an essential skill in calculus and is used extensively in mathematical analysis. The tangent to a curve is a straight line that touches the curve at a single point, and it is important in determining the rate of change of the function at that point. This article discusses the steps involved in the equation of the tangent to a curve, and provides some examples.

The first step in finding the equation of the tangent to a curve is to differentiate the function. The derivative of the function gives the slope of the tangent at any point on the curve. This is because the derivative of a function gives the rate of change of the function at any given point, and the slope of a tangent to a curve represents the rate of change of the function at that point.

Once the derivative of the function is found, it is evaluated at the point of interest to get the slope of the tangent at that point. This generally involves substituting the value of x at the point of interest into the derivative equation. The resulting value gives the slope of the tangent at that point.

The next step is to find the y-coordinate of the point of interest. This is usually done by substituting the value of x at the point of interest into the original function. The resulting value gives the y-coordinate of the point where the tangent will touch the curve.

With the slope and point established, the point-slope form of the equation of a straight line can be used to find the equation of the tangent. This equation gives the equation of the straight line that passes through the point of interest and has the slope that was calculated previously.

As an example, consider the function f(x) = x^2 – 4x + 3. The goal is to find the equation of the tangent to the curve at the point (2,-1).

The first step is to find the derivative of the function. The derivative of f(x) is 2x – 4.

The second step is to evaluate the derivative at the point of interest. When x = 2, the slope of the tangent is 0.

The third step is to find the y-coordinate of the point of interest. When x = 2, f(2) = 3.

The last step is to use the point-slope form of the equation of a straight line to find the equation of the tangent. Using the point (2,-1) and the slope 0, the equation of the tangent is y = -1.

In another example, consider the function g(x) = x^3 – 3x^2 + 2x. The goal is to find the equation of the tangent to the curve at the point (-1,6).

The first step is to find the derivative of g(x). The derivative is 3x^2 – 6x + 2.

The second step is to evaluate the derivative at the point of interest. When x = -1, the slope of the tangent is 17.

The third step is to find the y-coordinate of the point of interest. When x = -1, g(-1) = 6.

The last step is to use the point-slope form of the equation of a straight line to find the equation of the tangent. Using the point (-1,6) and the slope 17, the equation of the tangent is y = 17x + 23.

In summary, finding the equation of the tangent to a curve involves differentiating the function, evaluating the derivative at the point of interest, finding the y-coordinate of the point of interest, and using the point-slope form of the equation of a straight line to find the equation of the tangent. This skill is important in calculus and is used extensively in mathematical analysis.

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