Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 2/√x; P(4,1)
2. Intro to Derivatives
Tangent Lines and Derivatives
4:40 minutesProblem 31b
Textbook Question
Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)
Verified step by step guidance1
Step 1: Understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y_1 = m(x - x_1), where m is the slope of the tangent line, and (x_1, y_1) is the point of tangency.
Step 2: Identify the function f(x) = \sqrt{x} + 3 and the point P(1, 2). Here, x_1 = 1 and y_1 = 2.
Step 3: To find the slope m of the tangent line, calculate the derivative of the function f(x). The derivative f'(x) represents the slope of the tangent line at any point x.
Step 4: Differentiate f(x) = \sqrt{x} + 3. The derivative f'(x) = \frac{1}{2\sqrt{x}}. This is because the derivative of \sqrt{x} is \frac{1}{2\sqrt{x}}, and the derivative of a constant is 0.
Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at P. Substitute x = 1 into f'(x) to get m = f'(1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line Definition
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Mathematically, it represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function evaluated at that point.
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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) = √(x + 3), the derivative will provide the slope of the tangent line at point P.
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Point-Slope Form
The point-slope form of a linear equation is given by 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 a tangent line once the slope (derivative) and the point of tangency are known.
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