Problem 15 - University Calculus: Early Transcendentals

Question

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$h(t)=t^{3}, \quad\quad(2,8)$$

Step-by-Step Solution

Answer

The slope at (2,8) is 12, and the tangent line is y = 12t - 16.

Step 1: Understand the Problem

We need to find the slope of the tangent line to the function \(h(t) = t^3\) at the point \((2, 8)\). This requires calculating the derivative \(h'(t)\) to find the rate of change at \(t = 2\).

Step 2: Differentiate the Function

To find \(h'(t)\), the derivative of \(h(t) = t^3\), we use the power rule. The power rule states that \(\frac{d}{dt}[t^n] = n \cdot t^{n-1}\). Therefore, the derivative \(h'(t) = 3t^2\).

Step 3: Calculate the Slope at t = 2

Substitute \(t = 2\) into \(h'(t)\) to find the slope of the tangent line at this point. \[h'(2) = 3(2)^2 = 3 \cdot 4 = 12\]. So, the slope of the tangent line at \((2, 8)\) is 12.

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Substitute \(m = 12\), \(x_1 = 2\), and \(y_1 = 8\): \[y - 8 = 12(t - 2)\]. This simplifies to \(y = 12t - 16\).

Key Concepts

Tangent Line

A tangent line to a curve is a straight line that touches the curve at exactly one point, without crossing it at that point. This line gives a perfect snapshot of the curve's behavior at a specific point.
This is crucial when analyzing functions, as tangent lines can show how a function is behaving locally.

• The tangent line at a point gives the best linear approximation of the curve near that point.
• It is defined in terms of the slope, which quantifies its steepness or incline relative to the horizontal axis.
• In our exercise, finding the tangent line at the point \((2, 8)\) means determining its exact behavior on the curve \(h(t) = t^3\) right at that point.

Finding a tangent line involves two main steps: calculating the slope using derivatives and applying the point-slope form to find the equation. Understanding tangent lines is a building block in calculus, helping students to analyze complex functions by considering their linear approximations.

Derivative

In calculus, the derivative of a function measures how the function's output changes as its input changes. This is the concept of the rate of change, just like speed tells us how fast something is moving. Derivatives quantify how sensitive a function is to its inputs.

• For a function \(h(t)\), the derivative, denoted \(h'(t)\), represents the slope of the tangent line to the function at any given point \(t\).
• In our example, \(h'(t)\) tells us the rate at which \(h(t)\) changes as \(t\) changes.
• Finding a derivative involves algebraic techniques, often simplifying complex functions into linear approximations.

The derivative is central to calculus, enabling the study of otherwise intractable functions via simpler linear models. In the context of our exercise, computing \(h'(2)\) helps us find the slope and thus the equation of the tangent line at the specified point.

Power Rule

The power rule is a fundamental tool in differentiation, allowing us to find derivatives of polynomial functions quickly. The rule is straightforward and states: If you have a function \(t^n\), its derivative is \(n \, t^{n-1}\).
This is applied in our function \(h(t) = t^3\).

• Here, \(n = 3\), which makes our task simpler, as the derivative becomes \(3t^2\).
• The power rule automates what would otherwise be a complicated process of finding slopes manually.
• Because it works for any power \(n\), it's very useful, especially for students beginning calculus.

Using the power rule simplifies finding derivatives of polynomials, making differentiation almost mechanical. It's crucial for efficiently solving exercises like the one presented, where polynomial functions are involved.

Slope

The slope of a line is a measure of how steep the line is. In the context of curves, it represents the rate at which something changes. The slope, in general, is represented by \(m\), and in calculus, it's linked directly to the derivative.
For the tangent line, the slope is the derivative of the function evaluated at that particular point.

• In our example, the slope is found by evaluating the derivative, \(h'(t)\), at \(t = 2\).
• With the power rule, the derivative is \(3t^2\), and substituting \(t = 2\) gives \(3 \times 2^2 = 12\).
• This slope, \(12\), is crucial because it characterizes the tangent line and its angle relative to the curve \(h(t) = t^3\) at the point \((2, 8)\).

Understanding slopes, especially through the lens of derivatives, is fundamental in calculus as it ties directly into understanding the behavior of functions at specific points. The slope provides a numeric view of a curve's incline, crucial for engineering, physics, and mathematics alike.