Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = -3x² - 5x + 1; P(1,-7)

Verified step by step guidance

1. Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function \( f(x) \) at a point \( x = a \) is given by \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
2. Step 2: Identify the function \( f(x) = -3x^2 - 5x + 1 \) and the point \( P(1, -7) \). We need to find \( f'(1) \) to determine the slope of the tangent line at \( x = 1 \).
3. Step 3: Substitute \( a = 1 \) into the limit definition: \( f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} \).
4. Step 4: Calculate \( f(1) \) by substituting \( x = 1 \) into \( f(x) \): \( f(1) = -3(1)^2 - 5(1) + 1 \).
5. Step 5: Calculate \( f(1+h) \) by substituting \( x = 1+h \) into \( f(x) \): \( f(1+h) = -3(1+h)^2 - 5(1+h) + 1 \). Expand and simplify this expression.

Key Concepts

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.

Limit Definition of Derivative

The limit definition of the derivative states that the derivative f'(a) at a point a is the limit of the difference quotient as h approaches zero: f'(a) = lim(h→0) [(f(a+h) - f(a))/h]. This definition is fundamental for calculating the slope of the tangent line using the function's values at points near P.