Applications of Differentiation Revision Notes for HSC SSCE Mathematics Advanced

Applications of Differentiation

Introduction

Differentiation techniques can be applied to exponential functions in much the same way as other functions. One of the most important applications is curve sketching. When working with exponential functions, you may need to evaluate certain limits, which would typically be provided in exam questions.

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When working with exponential functions in exams, limit values are typically provided in the question. You should focus on understanding how to use these limits rather than proving them rigorously.

The graphs of $e^x$ and $e^{-x}$

These two graphs form the foundation of exponential functions. Understanding their properties is essential for working with more complex exponential expressions.

Properties of $y = e^x$

The function $y = e^x$ represents exponential growth. Key features include:

The graph passes through the point $(0, 1)$
As $x$ increases, $y$ increases rapidly without bound
As $x$ decreases, $y$ approaches zero (but never reaches it)
The $x$ -axis is a horizontal asymptote as $x \to -\infty$
The gradient at $(0, 1)$ is $1$

Sample values:

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$\frac{1}{e^2}$	$\frac{1}{e}$	$1$	$e$	$e^2$

Properties of $y = e^{-x}$

The function $y = e^{-x}$ represents exponential decay. Key features include:

The graph passes through the point $(0, 1)$
As $x$ increases, $y$ approaches zero
As $x$ decreases, $y$ increases without bound
The $x$ -axis is a horizontal asymptote as $x \to \infty$
The gradient at $(0, 1)$ is $-1$

Sample values:

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$e^2$	$e$	$1$	$\frac{1}{e}$	$\frac{1}{e^2}$

Relationship between the two graphs

Because $x$ is replaced by $-x$ in the second equation, the graphs of $y = e^x$ and $y = e^{-x}$ are reflections of each other in the $y$ -axis.

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The two curves intersect at $(0, 1)$ . At this point, the gradient of $y = e^x$ is $1$ and the gradient of $y = e^{-x}$ is $-1$ . This means the tangent lines are perpendicular at their point of intersection.

Real-world applications

The exponential decay function $y = e^{-x}$ is particularly important in applications. It describes many physical situations where a quantity "dies away exponentially", such as:

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Common Applications of Exponential Decay:

The decay of sound from a plucked string
Radioactive decay
Cooling of hot objects
Discharge of capacitors in electrical circuits

These applications all share the common feature of a quantity decreasing at a rate proportional to its current value.

Curve sketching with exponential functions

When sketching curves involving exponential functions, follow a systematic approach using these six steps:

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Six-Step Curve Sketching Method:

Write down the domain
Test whether the function is even, odd, or neither
Find any zeroes and examine the sign of the function
Examine behavior as $x \to \infty$ and $x \to -\infty$ , noting any asymptotes
Find stationary points and examine their nature
Find points of inflection and examine concavity

Following these steps systematically ensures you don't miss any important features of the curve.

Worked example: Sketching $y = xe^{-x}$

Let's apply these six steps to sketch the graph of $y = xe^{-x}$ .

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Worked Example: Sketching $y = xe^{-x}$ using the Six-Step Method

Step 1: Domain

The domain of $y = xe^{-x}$ is all real numbers.

Step 2: Even or odd function?

Testing $f(-x)$ :

$f(-x) = (-x)e^{-(-x)} = -xe^x$

This is neither $f(x)$ nor $-f(x)$ , so the function is neither even nor odd.

Step 3: Zeroes and sign

The only zero occurs when $x = 0$ .

From the table of signs:

$x$	$-1$	$0$	$1$
$y$	$-e$	$0$	$e^{-1}$
sign	$-$	$0$	$+$

The function is positive for $x > 0$ and negative for $x < 0$ .

Step 4: Asymptotic behavior

As $x \to \infty$ : $y \to 0$ (given), so the $x$ -axis is a horizontal asymptote on the right.

As $x \to -\infty$ : $y \to -\infty$

Step 5: Stationary points

Using the product rule with $u = x$ and $v = e^{-x}$ :

u' = 1 v' = -e^{-x}

Therefore:

f'(x) = vu' + uv' f'(x) = e^{-x}(1) + x(-e^{-x}) f'(x) = e^{-x} - xe^{-x} f'(x) = e^{-x}(1 - x)

Setting $f'(x) = 0$ :

Since $e^{-x}$ can never be zero, we need $1 - x = 0$ , giving $x = 1$ .

When $x = 1$ : $y = 1 \cdot e^{-1} = \frac{1}{e}$

So $\left(1, \frac{1}{e}\right)$ is the only stationary point.

Determining the nature of the stationary point:

Differentiating again using the product rule with $u = 1 - x$ and $v = e^{-x}$ :

u' = -1 v' = -e^{-x}

Therefore:

f''(x) = vu' + uv' f''(x) = e^{-x}(-1) + (1 - x)(-e^{-x}) f''(x) = -e^{-x} - (1 - x)e^{-x} f''(x) = e^{-x}(x - 2)

At $x = 1$ :

$f''(1) = e^{-1}(1 - 2) = -e^{-1} < 0$

Since $f''(1) < 0$ , the point $\left(1, e^{-1}\right)$ is a maximum turning point.

Step 6: Points of inflection

From $f''(x) = e^{-x}(x - 2)$ , we find $f''(x) = 0$ when $x = 2$ .

When $x = 2$ : $y = 2e^{-2}$

Testing concavity around $x = 2$ :

$x$	$0$	$2$	$3$
$f''(x)$	$-2$	$0$	$e^{-3}$
concavity	$\cap$	•	$\cup$

There is a point of inflection at $(2, 2e^{-2}) \approx (2, 0.27)$ .

The curve is concave down for $x < 2$ and concave up for $x > 2$ .

The complete sketch:

Graph transformations

You can use transformations to sketch related functions without repeating the entire curve-sketching process.

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Worked Example: Sketching $y = -xe^x$ using Transformations

Starting from the graph of $y = xe^{-x}$ , we can obtain $y = -xe^x$ by replacing $x$ with $-x$ .

Graphically, this transformation is a reflection in the $y$ -axis. The key features transform as follows:

Maximum at $\left(1, \frac{1}{e}\right)$ becomes maximum at $\left(-1, \frac{1}{e}\right)$
Inflection at $(2, 2e^{-2})$ becomes inflection at $(-2, 2e^{-2})$
Behavior as $x \to \infty$ becomes behavior as $x \to -\infty$ (and vice versa)

The reflected graph shows the same general shape but mirrored across the $y$ -axis.

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When using transformations, remember that replacing $x$ with $-x$ creates a reflection in the $y$ -axis, while replacing $y$ with $-y$ (or multiplying the entire function by $-1$ ) creates a reflection in the $x$ -axis.

Understanding limits involving exponential functions

When working with products of polynomials and exponential functions, the limiting behavior can be puzzling.

The limit of $xe^{-x}$ as $x \to \infty$

Consider the product $xe^{-x}$ when $x$ is large:

$x$ is a large number
$e^{-x}$ is a very small positive number

The product of a large number and a small number could theoretically be large, small, or anywhere in between.

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However, $e^{-x}$ decreases much faster than $x$ increases. We say that $e^{-x}$ dominates $x$ . The result is:

$\lim_{x \to \infty} xe^{-x} = 0$

This concept of exponential dominance is crucial for understanding limiting behavior.

This is shown in the following table:

$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$xe^{-x}$	$0$	$\frac{1}{e}$	$\frac{2}{e^2}$	$\frac{3}{e^3}$	$\frac{4}{e^4}$	$\frac{5}{e^5}$	$\frac{6}{e^6}$	$\frac{7}{e^7}$
approx	$0$	$0.37$	$0.27$	$0.15$	$0.073$	$0.034$	$0.015$	$0.006$

The limit of $xe^x$ as $x \to -\infty$

Similarly, when $x$ is a large negative number:

$x$ is a large negative number
$e^x$ is a very small positive number

Again, the exponential function dominates, and:

$\lim_{x \to -\infty} xe^x = 0$

This is shown in the following table:

$x$	$0$	$-1$	$-2$	$-3$	$-4$	$-5$	$-6$	$-7$
$xe^x$	$0$	$-\frac{1}{e}$	$-\frac{2}{e^2}$	$-\frac{3}{e^3}$	$-\frac{4}{e^4}$	$-\frac{5}{e^5}$	$-\frac{6}{e^6}$	$-\frac{7}{e^7}$
approx	$0$	$-0.37$	$-0.27$	$-0.15$	$-0.073$	$-0.034$	$-0.015$	$-0.006$

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Exam Tip:

In examination questions, limits such as these would normally be provided. You are not expected to prove them rigorously, but you should understand what they mean and how to use them in curve sketching.

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Key Points to Remember:

The graphs of $y = e^x$ and $y = e^{-x}$ are reflections of each other in the $y$ -axis and intersect perpendicularly at $(0, 1)$ .
When curve sketching with exponential functions, follow the systematic six-step approach: domain, even/odd, zeroes and sign, asymptotes, stationary points, and inflection points.
Use the product rule when differentiating products involving exponential functions.
Exponential functions dominate polynomial functions in limits, meaning $e^{-x}$ decreases faster than any polynomial increases.
Graph transformations (like reflection in the $y$ -axis) can be used to sketch related functions efficiently.

Applications of Differentiation (HSC SSCE Mathematics Advanced): Revision Notes