In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

f(x) = √(x + 1), (8, 3)

검증된 단계별 안내

To find the slope of the function's graph at the given point, we need to compute the derivative of the function f(x) = √(x + 1). The derivative, f'(x), represents the slope of the tangent line at any point x.

Use the chain rule to differentiate f(x) = (x + 1)^(1/2). The chain rule states that if you have a composite function g(h(x)), the derivative is g'(h(x)) * h'(x). Here, g(u) = u^(1/2) and h(x) = x + 1.

Differentiate g(u) = u^(1/2) to get g'(u) = (1/2)u^(-1/2). Differentiate h(x) = x + 1 to get h'(x) = 1. Therefore, f'(x) = (1/2)(x + 1)^(-1/2) * 1.

Evaluate the derivative at the given point x = 8 to find the slope of the tangent line. Substitute x = 8 into f'(x) to get f'(8) = (1/2)(8 + 1)^(-1/2).

Now, use the point-slope form of a line to find the equation of the tangent line. The point-slope form is y - y1 = m(x - x1), where m is the slope found in the previous step, and (x1, y1) is the given point (8, 3). Substitute these values into the equation to find the tangent line.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Derivative

The derivative of a function at a point provides the slope of the tangent line to the function's graph at that point. It is a fundamental concept in calculus that measures how a function changes as its input changes. For the function f(x) = √(x + 1), the derivative can be found using the power rule and chain rule.

Tangent Line

A tangent line to a curve at a given point is a straight line that just 'touches' the curve at that point. It has the same slope as the curve at that point, which is given by the derivative. The equation of the tangent line can be found using the point-slope form of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point of tangency.

Point-Slope Form

The point-slope form of a linear equation is used to find the equation of a line when you know the slope and a point on the line. It is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a specific point on the line. This form is particularly useful for writing the equation of a tangent line once the slope is determined from the derivative.

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.f(x) = √(x + 1), (8, 3)