Problem 7
Question
Find an equation of the tangent line to the curve at the given point. \( y = \sqrt{x} \), \( (1, 1) \)
Step-by-Step Solution
Verified Answer
The equation of the tangent line is \( y = \frac{1}{2}x + \frac{1}{2} \).
1Step 1: Find the Derivative
To find the equation of the tangent line, we need the slope at the given point, which requires finding the derivative of the function. For the function \( y = \sqrt{x} \), rewrite this as \( y = x^{1/2} \). The derivative \( y' \) is found using the power rule. Therefore, \( y' = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \).
2Step 2: Evaluate the Derivative at the Given Point
Substitute the x-coordinate of the point \((1,1)\) into the derivative to calculate the slope of the tangent line at that point. Substitute \( x = 1 \) into \( y' = \frac{1}{2\sqrt{x}} \) to get \( y' = \frac{1}{2\sqrt{1}} = \frac{1}{2} \). Thus, the slope of the tangent line at \((1,1)\) is \( \frac{1}{2} \).
3Step 3: Use the Point-Slope Form of a Line
With the slope \( m = \frac{1}{2} \) and the point \((1, 1)\), use the point-slope form, which is \( y - y_1 = m(x - x_1) \). Substitute \( m = \frac{1}{2} \), \( x_1 = 1 \), and \( y_1 = 1 \) to get the equation: \[ y - 1 = \frac{1}{2}(x - 1) \].
4Step 4: Simplify the Equation
Distribute \( \frac{1}{2} \) in the equation \( y - 1 = \frac{1}{2}(x - 1) \) to simplify: \[ y - 1 = \frac{1}{2}x - \frac{1}{2} \]. Then, add 1 to both sides to get the final equation: \[ y = \frac{1}{2}x + \frac{1}{2} \].
Key Concepts
Understanding DerivativeApplication of Power RulePoint-Slope Form InsightExploring Curves and Tangent Lines
Understanding Derivative
A derivative is a fundamental concept in calculus. It represents the rate at which a function is changing at any given point. Think of it as a measure of how steep a curve is at a particular spot. The derivative is crucial in finding the slope of tangent lines to curves. For a given function, the derivative provides a new function that describes these rates of change.
When faced with a curve, to find the tangent line, you need the slope at the specific point on the curve. This is accomplished by taking the derivative of the function that describes the curve. For example, the derivative of a function like \( y = x^{1/2} \) shows us how the function \( y \) changes as \( x \) changes. So, in essence, the derivative gives us the necessary slope to draw the tangent line.
When faced with a curve, to find the tangent line, you need the slope at the specific point on the curve. This is accomplished by taking the derivative of the function that describes the curve. For example, the derivative of a function like \( y = x^{1/2} \) shows us how the function \( y \) changes as \( x \) changes. So, in essence, the derivative gives us the necessary slope to draw the tangent line.
Application of Power Rule
The power rule is a basic derivative rule useful when differentiating functions with terms like \( x^n \). The rule states that if \( y = x^n \), then the derivative \( y' = nx^{n-1} \). This becomes extremely handy, as it simplifies the process of finding derivatives for polynomial functions and ones easily convertible to such a form.
In our example of finding a tangent line to \( y = \sqrt{x} \) at \( (1, 1) \), we first rewrite \( \sqrt{x} \) as \( x^{1/2} \). Then, applying the power rule gives us the derivative: \( y' = \frac{1}{2}x^{-1/2} \), or \( y' = \frac{1}{2\sqrt{x}} \). This derivative helps us find the slope of the tangent line at any given point, making it an essential tool in calculus.
In our example of finding a tangent line to \( y = \sqrt{x} \) at \( (1, 1) \), we first rewrite \( \sqrt{x} \) as \( x^{1/2} \). Then, applying the power rule gives us the derivative: \( y' = \frac{1}{2}x^{-1/2} \), or \( y' = \frac{1}{2\sqrt{x}} \). This derivative helps us find the slope of the tangent line at any given point, making it an essential tool in calculus.
Point-Slope Form Insight
The point-slope form of a line gives you a straightforward way to write the equation of a line if a point on the line and the slope are known. It is given by the formula:
In our specific problem, with the slope \( m = \frac{1}{2} \) and the point \( (1, 1) \), using the point-slope form allows us to write the equation of the tangent line directly. Substituting into the formula yields \( y - 1 = \frac{1}{2}(x - 1) \). This form is particularly beneficial as it directly integrates the calculation of the slope, making it quick and efficient for writing out the line's equation.
- \( y - y_1 = m(x - x_1) \)
In our specific problem, with the slope \( m = \frac{1}{2} \) and the point \( (1, 1) \), using the point-slope form allows us to write the equation of the tangent line directly. Substituting into the formula yields \( y - 1 = \frac{1}{2}(x - 1) \). This form is particularly beneficial as it directly integrates the calculation of the slope, making it quick and efficient for writing out the line's equation.
Exploring Curves and Tangent Lines
Curves are fascinating shapes that can have varying complexities. In calculus, analyzing these curves involves looking at their behavior and understanding how they change. This is where the concept of a tangent line comes into play. A tangent line is a straight line that touches the curve at exactly one point and matches the direction the curve is heading at that precise spot.
To find this tangent line's equation, the curve's derivative is used. For the curve given by \( y = \sqrt{x} \), the tangent line at the point \( (1, 1) \) provides a snapshot of how the curve behaves at that particular point. By determining the slope at \( (1, 1) \), using the derivative \( y' = \frac{1}{2\sqrt{x}} \), you understand more about the curve's nature where the line touches. Thus, tangent lines play a crucial role in visualizing and comprehending the properties of curves in a more tangible way.
To find this tangent line's equation, the curve's derivative is used. For the curve given by \( y = \sqrt{x} \), the tangent line at the point \( (1, 1) \) provides a snapshot of how the curve behaves at that particular point. By determining the slope at \( (1, 1) \), using the derivative \( y' = \frac{1}{2\sqrt{x}} \), you understand more about the curve's nature where the line touches. Thus, tangent lines play a crucial role in visualizing and comprehending the properties of curves in a more tangible way.
Other exercises in this chapter
Problem 6
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