Problem 4 - Calculus and its applications

Question

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=-3 x^{2}$$

Step-by-Step Solution

Answer

The slopes of the tangent lines at \( x = -2, 0, \) and \( 1 \) are \( 12, 0, \) and \( -6 \) respectively.

Step 1: Graph the Function

Plot the function \( f(x) = -3x^2 \) on a coordinate plane. This function is a downward-opening parabola, with its vertex at the origin \((0,0)\). The graph is symmetrical about the y-axis.

Step 2: Draw Tangent Lines

To draw tangent lines at the points where \( x = -2, 0, \) and \( 1 \), determine their corresponding y-values on the graph. At \( x = -2 \), \( y = -3(-2)^2 = -12 \). At \( x = 0 \), \( y = -3(0)^2 = 0 \). At \( x = 1 \), \( y = -3(1)^2 = -3 \). Draw tangent lines that touch the graph at these points.

Step 3: Calculate the Derivative

Find the derivative \( f'(x) \) using the limit definition: \ \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. \ Substitute \( f(x) = -3x^2 \) giving: \ \[ f'(x) = \lim_{h \to 0} \frac{-3(x+h)^2 + 3x^2}{h} \]. \ Simplify: \ \[ f'(x) = \lim_{h \to 0} \frac{-3(x^2 + 2xh + h^2) + 3x^2}{h} = \lim_{h \to 0} \frac{-3x^2 - 6xh - 3h^2 + 3x^2}{h} \]. \ Simplifying further, \ \[ f'(x) = \lim_{h \to 0} \frac{-6xh - 3h^2}{h} = \lim_{h \to 0} (-6x - 3h) \]. \ Conclude: \[ f'(x) = -6x \].

Step 4: Find Derivatives at Specific Points

Evaluate \( f'(x) \) for \( x = -2, 0, \) and \( 1 \). First, for \( x = -2 \), \( f'(-2) = -6(-2) = 12 \). Next, for \( x = 0 \), \( f'(0) = -6(0) = 0 \). Finally, for \( x = 1 \), \( f'(1) = -6(1) = -6 \). Compare these slopes to the tangent lines drawn in Step 2.

Key Concepts

Derivative

The derivative is a fundamental tool in calculus used to determine how a function changes at any given point. In essence, it tells you the "rate of change" or "slope" of the function. For example, consider the function \( f(x) = -3x^2 \). The derivative, represented as \( f'(x) \), gives us the slope of the tangent line to the graph of the function at any point \( x \).Key points about derivatives:

• They show how fast or slow a function is changing.
• Positive derivatives indicate an increasing function, while negative ones point to a decreasing function.
• For constant functions, the derivative is zero since there is no change.

In practice, finding the derivative involves calculus techniques such as the limit definition, which lets you calculate the derivative at a point by considering how the function behaves as you make infinitesimally small changes to \( x \).

Limit Definition

The limit definition of the derivative is a crucial concept in calculus. It allows us to compute the derivative of a function at a specific point. The basic idea is to find the slope of the tangent line to the curve at that point.Mathematically, the limit definition is expressed as:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]This expression means you take two points close to each other on the curve \( f(x) \) and look at the slope between them as the distance \( h \) approaches zero.This process involves:

• Choosing a small \( h \) value
• Calculating \( f(x+h) \) and \( f(x) \)
• Subtracting \( f(x) \) from \( f(x+h) \)
• Dividing by \( h \) and taking the limit as \( h \) goes to zero

In our example with \( f(x) = -3x^2 \), using the limit definition gives us a derivative \( f'(x) = -6x \). This shows that the change in \( f \) is proportional to \( x \), scaled by \(-6\).

Tangent Lines

Tangent lines play an essential role in understanding how functions behave. A tangent line to a graph at a given point is a straight line that "just touches" the curve at that point. It represents the instantaneous rate of change of the function, or the derivative, at that specific point.Here's how to find tangent lines at certain points:

• Identify the point where you want the tangent.
• Compute the derivative at that point, which provides the slope of the tangent line.
• Use 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 point of tangency.

For the function \( f(x) = -3x^2 \), we drew tangent lines at \( x = -2, 0, \) and \( 1\). Calculated slopes (12, 0, and -6 respectively) match the graph, verifying the validity of both our derivative and visual checks.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. It is the graph of a quadratic function, such as \( f(x) = ax^2 + bx + c \). In our example, the function \( f(x) = -3x^2 \) describes a downward-opening parabola, with its vertex at the origin \((0,0)\).Characteristics of a parabola:

• The vertex is the highest or lowest point of the curve, depending on its orientation.
• If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• It is symmetrical about its line of symmetry, often the y-axis if the function is simple.

Understanding parabolas is crucial since they often appear in physics and engineering contexts when analyzing patterns of motion and forces. In calculus, their simple derivatives and clear graphical characteristics make them excellent examples to study tangents and dynamic changes.