Problem 62
Question
For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=x^{2}-3 $$
Step-by-Step Solution
Verified Answer
The tangent line is horizontal at the point \((0, -3)\).
1Step 1: Identify the Function and Derivative
The function given is \( y = x^2 - 3 \). To find the points where the tangent line is horizontal, we need to first find the derivative of the function. The derivative of the function \( y = x^2 - 3 \) is obtained by differentiating with respect to \( x \).
2Step 2: Differentiate the Function
Differentiate the function \( y = x^2 - 3 \). Applying the power rule of differentiation, we get the derivative as \( \frac{dy}{dx} = 2x \).
3Step 3: Find Where the Derivative is Zero
A horizontal tangent line implies the derivative is zero. Set \( \frac{dy}{dx} = 2x = 0 \) and solve for \( x \).
4Step 4: Solve the Equation
Solving the equation \( 2x = 0 \) gives \( x = 0 \). Thus, the point where the derivative is zero is when \( x = 0 \).
5Step 5: Determine y-coordinate of the Point
Substitute \( x = 0 \) back into the original function \( y = x^2 - 3 \) to find the corresponding \( y \)-coordinate: \( y = 0^2 - 3 = -3 \).
6Step 6: Conclusion on Points of Horizontal Tangency
The point on the graph where the tangent line is horizontal is \((0, -3)\).
Key Concepts
Horizontal Tangent LineDerivativePower Rule of DifferentiationGraph of a Function
Horizontal Tangent Line
A horizontal tangent line on a graph is a straight line that touches the curve at a single point without crossing it, and is perfectly flat along that section. This occurs where the slope of the tangent is zero.
To determine where a tangent line is horizontal, you need to find where the derivative of the function equals zero. At these points, the curve pauses its upward or downward trend momentarily.
In the exercise, this approach determined that at the point \(x = 0\), there is a horizontal tangent with the function \(y = x^2 - 3\).
To determine where a tangent line is horizontal, you need to find where the derivative of the function equals zero. At these points, the curve pauses its upward or downward trend momentarily.
- Step 1: Find the derivative of the function.
- Step 2: Set the derivative equal to zero and solve for the variable.
In the exercise, this approach determined that at the point \(x = 0\), there is a horizontal tangent with the function \(y = x^2 - 3\).
Derivative
The derivative of a function represents its rate of change. In simpler terms, it tells you how fast or slow a function's value is changing at any given \(x\) value.
Understanding derivatives is crucial in calculus as it lays the foundation for concepts like motion, optimization, and finding tangents.
For the given function \(y = x^2 - 3\), the derivative \(\frac{dy}{dx} = 2x\) tells us that the slope is zero when \(x = 0\).
Understanding derivatives is crucial in calculus as it lays the foundation for concepts like motion, optimization, and finding tangents.
- The function's derivative indicates the slope of the tangent line at any point on the curve.
- Setting the derivative to zero can help pinpoint points of horizontal tangency.
For the given function \(y = x^2 - 3\), the derivative \(\frac{dy}{dx} = 2x\) tells us that the slope is zero when \(x = 0\).
Power Rule of Differentiation
The power rule is a straightforward method for finding the derivative of any term that is a power of \(x\). It simplifies the process significantly.
If you have a term like \(x^n\), where \(n\) is a constant, the power rule states that its derivative is \(nx^{n-1}\).
In the given function \(y = x^2 - 3\), using the power rule, the derivative \(\frac{dy}{dx} = 2x\) is easily computed, making it quick to analyze the curve's nature.
If you have a term like \(x^n\), where \(n\) is a constant, the power rule states that its derivative is \(nx^{n-1}\).
- For example, the derivative of \(x^2\) is \(2x\) using the power rule.
- Subtract 1 from the original exponent to adjust the power.
In the given function \(y = x^2 - 3\), using the power rule, the derivative \(\frac{dy}{dx} = 2x\) is easily computed, making it quick to analyze the curve's nature.
Graph of a Function
Every algebraic function has a graphical representation which shows how the function behaves in various \(x\) values. This visual depiction helps grasp concepts like roots, peaks, and points of horizontal tangency easily.
The graph of \(y = x^2 - 3\) is a parabola that opens upwards, shifted down by 3 units. At the vertex, \(x = 0\), the tangent line is horizontal, represented as \(y = -3\).
By understanding the function's graph, we not only identify the horizontal tangent at \(x = 0\) but also develop insights into the entire curve's nature.
The graph of \(y = x^2 - 3\) is a parabola that opens upwards, shifted down by 3 units. At the vertex, \(x = 0\), the tangent line is horizontal, represented as \(y = -3\).
- Understand how changes in function affect its graph's shape and position.
- Recognize that a tangent's slope helps determine curve behaviour at specific points.
By understanding the function's graph, we not only identify the horizontal tangent at \(x = 0\) but also develop insights into the entire curve's nature.
Other exercises in this chapter
Problem 61
Consider $$ g(x)=\left(\frac{6 x+1}{2 x-5}\right)^{2} $$ a) Find \(g^{\prime}(x)\) using the Extended Power Rule. b) Note that \(g(x)=\frac{36 x^{2}+12 x+1}{4 x
View solution Problem 61
Is the function given by $$ G(x)=\left\\{\begin{array}{ll} \frac{x^{2}-3 x-4}{x-4}, & \text { for } x
View solution Problem 62
Find \(y^{\prime \prime \prime}\) for each function. $$ y=\frac{1}{\sqrt{2 x+1}} $$
View solution Problem 62
Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ \begin{array}{l} G(x)=\left\\{\begin{array}{ll}
View solution