Problem 11 - Single Variable Calculus

Question

(a) use Equation (1) to find the slope of the secant line passing through the points \((a, f(a))\) and \((a+h, f(a+h)) ;\) (b) use the results of part (a) and Equafion (2) to find the slope of the tangent line at the point \((a, f(a)) ;\) and \((\mathrm{c})\) find an equation of the tangent line to the graph of \(f\) at the point \((a, f(a))\). \(f(x)=x^{3}\) \((2,8)\)

Step-by-Step Solution

Answer

(a) The slope of the secant line is \(3a^2 + 3ah + h^2\). (b) The slope of the tangent line at \((2,8)\) is \(12\). (c) The equation of the tangent line to the graph of \(f\) at the point \((2, 8)\) is \(y = 12x - 16\).

Step 1: Find the slope of the secant line

To find the slope of the secant line using Equation (1), we need to determine the difference quotient: \[ \frac{f(a+h) - f(a)}{h} \] We know that \(f(x) = x^3\). Therefore, to calculate the slope, substitute \(f(a+h)\) and \(f(a)\) with the function's expression: \[ \frac{(a+h)^3 - a^3}{h} \]

Step 2: Simplify the difference quotient

Now, we need to simplify the expression: \[ \frac{(a+h)^3 - a^3}{h} = \frac{a^3 + 3a^2h + 3ah^2 + h^3 - a^3}{h} \] We can cancel the \(a^3\) terms and then factor out an \(h\) from the remaining terms: \[ \frac{3a^2h + 3ah^2 + h^3}{h} = 3a^2 + 3ah + h^2 \] So, the slope of the secant line passing through the points \((a, f(a))\) and \((a+h, f(a+h))\) is \(3a^2 + 3ah + h^2\).

Step 3: Find the slope of the tangent line

To find the slope of the tangent line, we will use Equation (2). This means we need to take the limit of the slope of the secant line as \(h\) approaches 0: \[ \lim_{h \to 0} (3a^2 + 3ah + h^2) \] As \(h\) approaches 0, the terms with \(h\) will become 0: \[ \lim_{h \to 0} (3a^2 + 3ah + h^2) = 3a^2 \] Thus, the slope of the tangent line at \((a, f(a))\) is \(3a^2\).

Step 4: Find the slope at the given point

Now that we have the slope of the tangent line, we can find its value at the given point. For the point \((2,8)\), the value of \(a\) is 2. So, the slope of the tangent line at \((2, 8)\) is: \[ 3a^2 = 3(2)^2 = 3(4) = 12 \]

Step 5: Find the equation of the tangent line

To find the equation of the tangent line, we will use the point-slope form: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1) = (2, 8)\) and the slope \(m = 12\). Substituting the values, we get: \[ y - 8 = 12(x - 2) \] Now, simplify the equation: \[ y - 8 = 12x - 24 \] Add 8 to both sides to isolate y: \[ y = 12x - 16 \] So, the equation of the tangent line to the graph of \(f\) at the point \((2, 8)\) is \(y = 12x - 16\).

Key Concepts

Slope of Secant Line

Understanding the slope of a secant line is a foundational concept in calculus. It represents the average rate of change of the function between two points and is found using the difference quotient. For instance, consider the function f(x) = x^3 and two points on this curve, \( (a, f(a)) \) and \( (a+h, f(a+h)) \). The slope of the secant line that passes through these points is determined by the ratio
\[ \frac{f(a+h) - f(a)}{h} \]
which simplifies to
\[ 3a^2 + 3ah + h^2 \]
This expression gives us a numerical value that represents the average slope between two points on the curve. The value of \(h\) represents the horizontal distance between the two points, and as \(h\) gets smaller, the secant line approaches the tangent line at point \(a\).

Difference Quotient

The difference quotient is a crucial tool in calculus for measuring the rate of change of a function. It is expressed as
\[ \frac{f(a+h) - f(a)}{h} \]
and is the core of the derivative concept. For the cubic function \(f(x) = x^3\), the difference quotient becomes
\[ \frac{(a+h)^3 - a^3}{h} \]
Upon simplification, this quotient reveals the nature of change within the function for an increment \(h\). During the simplification process, we expand the cubic term, cancel like terms, and then factor out the \(h\). The resulting expression tells us how the function's output changes in response to a small change in input, providing a powerful insight into the function's behavior at a particular point.

Slope of Tangent Line

The slope of the tangent line at a specific point on a curve offers a precise measure of the function's rate of change at that point. It is what the slope of the secant line becomes as the two points on the curve get infinitesimally close. In calculus, we find this slope using the limit definition of derivative. Taking the limit of the difference quotient,
\[ \lim_{h \to 0} (3a^2 + 3ah + h^2) \]
allows us to eliminate the \(h\)-dependent terms and find that the slope of the tangent line to \(f(x) = x^3\) at the point \(a\) is
\[ 3a^2 \]
This value is derived by letting \(h\) approach zero, signifying that we are looking at the rate of change at a singular point rather than over an interval.

Limit Definition of Derivative

The limit definition of the derivative stands at the heart of differential calculus. It defines the derivative of a function at a point as the limit of the difference quotient as \(h\) approaches zero:
\[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]
This concept transforms the average rate of change (the slope of the secant line) into the instantaneous rate of change (the slope of the tangent line). Applying this to \(f(x) = x^3\), we get the derivative \(f'(x) = 3x^2\) by evaluating the limit
\[ \lim_{h \to 0} (3a^2 + 3ah + h^2) = 3a^2 \]
This process makes use of the foundational limit operations that underpin calculus and provides a generalized approach for finding derivatives of various functions.

Point-Slope Form Equation

Forming the Line Tangent to Curves

After determining the slope of the tangent line, we can develop the equation of the line tangent to a curve at a specific point using the point-slope form equation. It is stated as
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point of tangency and \(m\) is the slope of the tangent line. For \(f(x) = x^3\) at the point \((2,8)\), with a slope of 12, the tangent line equation is structured as
\[ y - 8 = 12(x - 2) \]
This equation leads to the linear expression
\[ y = 12x - 16 \]
which describes the line that just grazes the curve at that point, embodying the concept of the derivative geometrically.