Math Problem Statement

Find the equation of the tangent line to the curve at the point (1,𝑟(1))(1,r(1)).

𝑦=

Solution

To find the equation of the tangent line to the curve at the point (1,r(1))(1, r(1)), we need to follow these steps:

Step 1: Identify the curve

You have not provided the equation of the curve y=r(x)y = r(x). Could you clarify what function r(x)r(x) is? For example, is it a specific function like r(x)=x2r(x) = x^2 or something else?

Step 2: Use the formula for the equation of the tangent line

The general form of the equation of the tangent line at a point (x1,y1)(x_1, y_1) on the curve is:

yy1=m(xx1)y - y_1 = m(x - x_1)

where:

  • (x1,y1)=(1,r(1))(x_1, y_1) = (1, r(1))
  • mm is the slope of the tangent line, which is the derivative r(x)r'(x) evaluated at x=1x = 1.

Step 3: Find the derivative

To find the slope of the tangent line, we need to compute the derivative r(x)r'(x). If you can provide the equation for r(x)r(x), I can help you find the derivative.

Once I have the derivative, I can plug in x=1x = 1 to get the slope m=r(1)m = r'(1).

Step 4: Write the equation of the tangent line

Using the values m=r(1)m = r'(1) and y1=r(1)y_1 = r(1), I will plug them into the tangent line equation to give you the final result.

Please provide the function r(x)r(x) so I can continue!

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Tangent Lines

Formulas

Equation of the tangent line: y - y1 = m(x - x1)
Derivative of a function: r'(x)

Theorems

Derivative as the slope of the tangent line

Suitable Grade Level

Grades 11-12