Question

Напишите уравнение касательной к графику функции в точке

Answer 1

Відповідь:

ответ на фото

Answer 2

Finding the Equation of the Tangent Line

To find the equation of the tangent line to a function at a specific point, we can use the derivative of the function. The equation of the tangent line at a point (x1, y1) on the graph of the function y = f(x) is given by:

y - y1 = m(x - x1)

Where m is the slope of the tangent line, and (x1, y1) is the point of tangency.

Calculating the Slope

The slope of the tangent line, m, can be calculated using the derivative of the function. If the function is given by y = f(x), then the derivative f'(x) gives the slope of the tangent line at any point x.

Example Calculation

Let's consider an example where the function is given by y = x^2, and we want to find the equation of the tangent line at the point (2, 4).

1. Calculate the Derivative: The derivative of the function y = x^2 is given by f'(x) = 2x.

2. Calculate the Slope at the Given Point: Substituting x = 2 into the derivative, we get f'(2) = 2*2 = 4. This is the slope of the tangent line at the point (2, 4).

3. Write the Equation of the Tangent Line: Using the point-slope form, the equation of the tangent line is: y - 4 = 4(x - 2)

Simplifying, we get: y - 4 = 4x - 8 y = 4x - 4

So, the equation of the tangent line to the function y = x^2 at the point (2, 4) is y = 4x - 4.

Conclusion

In conclusion, to find the equation of the tangent line to a function at a specific point, we use the derivative of the function to calculate the slope of the tangent line and then apply the point-slope form to write the equation of the tangent line.