Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 3x² - 4x; P(1, -1)

검증된 단계별 안내

Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function f at a point x is given by the limit: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).

Step 2: Substitute the given function f(x) = 3x^2 - 4x into the derivative definition. Calculate f(x+h) by replacing x with (x+h) in the function: f(x+h) = 3(x+h)^2 - 4(x+h).

Step 3: Expand the expression for f(x+h): f(x+h) = 3(x^2 + 2xh + h^2) - 4x - 4h = 3x^2 + 6xh + 3h^2 - 4x - 4h.

Step 4: Substitute f(x) and f(x+h) into the derivative formula: \(\frac{f(x+h) - f(x)}{h}\) = \(\frac{(3x^2 + 6xh + 3h^2 - 4x - 4h) - (3x^2 - 4x)}{h}\).

Step 5: Simplify the expression: \(\frac{6xh + 3h^2 - 4h}{h}\) = 6x + 3h - 4. Take the limit as h approaches 0 to find f'(x) = 6x - 4. Evaluate f'(1) to find the slope of the tangent line at P(1, -1).

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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 crucial for understanding how the function behaves locally.

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) will provide the slope of the tangent line at point P.

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 useful for constructing the equation of the tangent line once the slope has been determined.

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.f(x) = 3x2 - 4x; P(1, -1)