If f′(−2) = 7, find an equation of the line tangent to the graph of f at the point (−2,4).

Verified step by step guidance

1. Step 1: Recall that the equation of a line in point-slope form is given by y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
2. Step 2: Identify the slope of the tangent line. Since f'(-2) = 7, the slope of the tangent line at x = -2 is 7.
3. Step 3: Identify the point on the graph where the tangent line touches. The problem states that the point is (-2, 4).
4. Step 4: Substitute the slope (m = 7) and the point (-2, 4) into the point-slope form equation: y - 4 = 7(x + 2).
5. Step 5: Simplify the equation if needed to express it in a different form, such as slope-intercept form (y = mx + b), by distributing and rearranging terms.

Key Concepts

Derivative

The derivative of a function at a given point represents the slope of the tangent line to the graph of the function at that point. In this case, f′(−2) = 7 indicates that the slope of the tangent line at x = -2 is 7.

Point-Slope Form

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a tangent line when a point and slope are known.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The equation of the tangent line can be derived using the slope from the derivative and the coordinates of the point on the curve.