Math Problem Statement

Find the equation of the tangent to the curve y = 3x^2 - 4 at x = 2.

Solution

To find the equation of the tangent line to the curve y=3x24y = 3x^2 - 4 at x=2x = 2, we need to follow these steps:

  1. Find the derivative of the function to get the slope of the tangent line.

    The given equation is y=3x24y = 3x^2 - 4. The derivative dydx\frac{dy}{dx} represents the slope of the tangent line at any point xx.

    dydx=ddx(3x24)=6x\frac{dy}{dx} = \frac{d}{dx}(3x^2 - 4) = 6x

  2. Evaluate the slope at x=2x = 2.

    Substituting x=2x = 2 into the derivative:

    slope=6(2)=12\text{slope} = 6(2) = 12

    So, the slope of the tangent line at x=2x = 2 is 12.

  3. Find the y-coordinate at x=2x = 2.

    To get the point where the tangent touches the curve, substitute x=2x = 2 into the original equation:

    y=3(2)24=124=8y = 3(2)^2 - 4 = 12 - 4 = 8

    Thus, the point of tangency is (2,8)(2, 8).

  4. Use the point-slope form of the equation of a line.

    The point-slope form is given by:

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

    where mm is the slope, and (x1,y1)(x_1, y_1) is the point on the line. Substituting m=12m = 12, x1=2x_1 = 2, and y1=8y_1 = 8:

    y8=12(x2)y - 8 = 12(x - 2)

    Simplifying this equation:

    y8=12x24y - 8 = 12x - 24 y=12x16y = 12x - 16

So, the equation of the tangent line is y=12x16y = 12x - 16.

Would you like any further details or explanations?

Here are some related questions you might find useful:

  1. How do you find the derivative of more complex functions like trigonometric or exponential functions?
  2. What is the geometric interpretation of the derivative in terms of slopes and tangents?
  3. Can the tangent line ever be vertical? Under what conditions?
  4. How does one find the normal line to a curve at a given point?
  5. What happens to the tangent line as xx \to \infty for polynomials of higher degrees?

Tip: When finding the tangent line, always ensure you evaluate both the slope and the point of tangency correctly for accurate results.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Quadratic Functions
Tangent Lines

Formulas

Derivative of y = 3x^2 - 4 is dy/dx = 6x
Point-slope form of a line: y - y1 = m(x - x1)

Theorems

Derivative provides the slope of the tangent to the curve at any given point.

Suitable Grade Level

Grades 10-12