Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.
f(x) = 1/x; a= -5

검증된 단계별 안내

Step 1: Understand that the equation of a tangent line to the graph of a function at a point (a, f(a)) is given by the formula y - f(a) = f'(a)(x - a), where f'(a) is the derivative of the function evaluated at x = a.

Step 2: Find the derivative of the function f(x) = 1/x. The derivative, f'(x), can be found using the power rule. Rewrite f(x) as x^(-1) and differentiate to get f'(x) = -1/x^2.

Step 3: Evaluate the derivative at the given point a = -5. Substitute x = -5 into f'(x) to find f'(-5) = -1/(-5)^2.

Step 4: Calculate f(a) by substituting a = -5 into the original function f(x) = 1/x. This gives f(-5) = 1/(-5).

Step 5: Substitute f(a) and f'(a) into the tangent line equation y - f(a) = f'(a)(x - a) to find the equation of the tangent line at the point (a, f(a)).

비슷한 문제에 대한 검증된 영상 답변:

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

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 the tangent line represents the instantaneous rate of change of the function at that point, which is determined by the derivative of the function.

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) = 1/x, the derivative can be found using the power rule or quotient rule.

Point-Slope Form

The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as 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 finding the equation of the tangent line once the slope and point are known.

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.f(x) = 1/x; a= -5