Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 2x + 1; P(0,1)

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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 the limit \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).

Step 2: Identify the function \( f(x) = 2x + 1 \) and the point \( P(0, 1) \). Here, \( a = 0 \) since we are finding the slope of the tangent line at \( x = 0 \).

Step 3: Substitute \( a = 0 \) into the derivative formula: \( f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \).

Step 4: Calculate \( f(0+h) \) and \( f(0) \). Since \( f(x) = 2x + 1 \), we have \( f(0+h) = 2(0+h) + 1 = 2h + 1 \) and \( f(0) = 2(0) + 1 = 1 \).

Step 5: Substitute these values into the limit expression: \( f'(0) = \lim_{h \to 0} \frac{(2h + 1) - 1}{h} = \lim_{h \to 0} \frac{2h}{h} \). Simplify the expression to find the slope of the tangent line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. For the function f(x) = 2x + 1, the derivative can be computed to find the slope of the tangent line at any point.

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 writing the equation of the tangent line once the slope has been determined.

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Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

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21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

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