Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = -7x; P(-1,7)
2. Intro to Derivatives
Tangent Lines and Derivatives
3:39 minutesProblem 3.1.26a
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 1/x; P (1,1)
Verified step by step guidance1
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 = a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function and the point of interest. Here, the function is \( f(x) = \frac{1}{x} \) and the point P is (1, 1). We need to find \( f'(1) \).
Step 3: Substitute \( f(x) = \frac{1}{x} \) into the derivative formula. This gives: \( f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{\frac{1}{1+h} - 1}{h} \).
Step 4: Simplify the expression inside the limit. Start by finding a common denominator for the terms in the numerator: \( \frac{1}{1+h} - 1 = \frac{1 - (1+h)}{1+h} = \frac{-h}{1+h} \).
Step 5: Substitute the simplified expression back into the limit: \( f'(1) = \lim_{h \to 0} \frac{-h}{h(1+h)} \). Simplify this to \( \lim_{h \to 0} \frac{-1}{1+h} \) and evaluate the limit as \( h \to 0 \).
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Video duration:
3mKey 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.
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Slopes of Tangent Lines
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) = 1/x, finding the derivative will provide the slope of the tangent line at point P.
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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, as it provides a precise method to determine the instantaneous rate of change at a specific point.
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