Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)

검증된 단계별 안내

Step 1: Understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y_1 = m(x - x_1), where m is the slope of the tangent line, and (x_1, y_1) is the point of tangency.

Step 2: Identify the function f(x) = \(\sqrt{x}\) + 3 and the point P(1, 2). Here, x_1 = 1 and y_1 = 2.

Step 3: To find the slope m of the tangent line, calculate the derivative of the function f(x). The derivative f'(x) represents the slope of the tangent line at any point x.

Step 4: Differentiate f(x) = \(\sqrt{x}\) + 3. The derivative f'(x) = \(\frac{1}{2\sqrt{x}\)}. This is because the derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}\)}, and the derivative of a constant is 0.

Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at P. Substitute x = 1 into f'(x) to get m = f'(1).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Tangent Line Definition

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Mathematically, it represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function evaluated at that point.

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is calculated as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = √(x + 3), the derivative will provide the slope of the tangent line at point P.

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 once the slope (derivative) and the point of tangency are known.

Equations of tangent lines by definition (2)b. Determine an equation of the tangent line at P.f(x) = √x+3; P (1,2)