If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at ​ x=1, 2, and 3​​​.

Verified step by step guidance

1. Step 1: Understand that the derivative of a function, f′(x), represents the slope of the tangent line to the curve y=f(x) at any point x.
2. Step 2: Recognize that you are given f′(x) = 3x + 2, which is the expression for the slope of the tangent line at any point x on the curve.
3. Step 3: To find the slope of the tangent line at x = 1, substitute x = 1 into the derivative: f′(1) = 3(1) + 2.
4. Step 4: To find the slope of the tangent line at x = 2, substitute x = 2 into the derivative: f′(2) = 3(2) + 2.
5. Step 5: To find the slope of the tangent line at x = 3, substitute x = 3 into the derivative: f′(3) = 3(3) + 2.

Key Concepts

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is represented as f'(x) and provides the slope of the tangent line to the curve at that specific point. In this question, f'(x) = 3x + 2 indicates how the slope varies with x.

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 slope of this line is equal to the derivative of the function at that point. To find the slope of the tangent line at x = 1, 2, and 3, we evaluate the derivative at these x-values.

Evaluating Functions

Evaluating a function involves substituting a specific value into the function to find its output. In this context, we will substitute x = 1, 2, and 3 into the derivative f'(x) = 3x + 2 to find the slopes of the tangent lines at these points. This process is essential for determining how the function behaves at specific locations.