### Solving Differentials Problems

The following problems involve the concept of the Differential of a Function. I will introduce the Differential via it's geometric interpretation and then formulate it. For a function $y=f(x)$ it will be shown that Differentials can be used to estimate the change of $y-$values as a function of a change in $x$-values. We will use Differentials to solve three types of problems. Differentials will be used to

$\ \ \ \ \$ I.) estimate the value of a numerical expression.
$\ \ \ \$ II.) approximate a relatively complicated functional expression with a simpler polynomial expression.
$\ \ \$ III.) estimate the propagation of percentage errors.

Let's begin with the graph of a function $y=f(x)$ and consider $x-$values changing from $x$ to $x+ \Delta x$, where $\Delta x$ will be used to represent a small positive or negative change in $x$. I will refer to $x$ as the "starting" $x-$value and to $x + \Delta x$ as the second $x-$value. Draw a tangent line to this graph at $x$.

Since $x-$values change from $x$ to $x+ \Delta x$, the corresponding $y-$values will change from $f(x)$ to $f(x + \Delta x)$. Define this exact change in $y$-values to be $\Delta y$, where $$\Delta y = f(x + \Delta x) - f(x)$$ We can now geometrically define this so-called Differential of $f(x)$. We will denote the Differential by $dy$. It is the HEIGHT of the designated right triangle formed by $x$, $x+ \Delta x$, and the tangent line to the graph of $y=f(x)$ at $x$ in the following diagram:

Let's find a formula for $dy$. Recall from algebra that the SLOPE of this tangent line at $x$ is $$m = \displaystyle{ rise \over run } = \displaystyle{ dy \over \Delta x }$$ Recall also that the SLOPE of this tangent line at $x$ is the derivative $$m=f'(x)$$ Setting these slopes equal to each other gives us $$\displaystyle{ dy \over \Delta x } = f'(x)$$

so that the Differential of Function $f$ at $x$ is

$$dy = f'(x) \ \Delta x$$

Let's now verify that the Exact Change of $\ y=f(x) \$ is approximately equal to the Differential of $\ y=f(x) \$ for "small" $\Delta x$, i.e.,

$$\Delta y \approx dy$$

for small $\Delta x$. Recall that the Derivative of $f$ at $x$ is $$\displaystyle{ \lim_{ \Delta x \to 0 } { f(x + \Delta x) - f(x) \over \Delta x } } = f'(x) \ \ \ \ \longrightarrow$$ $$\displaystyle{ { f(x + \Delta x) - f(x) \over \Delta x } } \approx f'(x) \ \ \ \ \longrightarrow$$
for "small" $\Delta x$
$$\displaystyle{ { \Delta y \over \Delta x } } \approx f'(x) \ \ \ \ \longrightarrow$$ $$\Delta y \approx f'(x) \ \Delta x \ \ \ \ \longrightarrow$$ $$\Delta y \approx dy$$
for "small" $\Delta x$.

Here is a summary of Differentials facts.

$\ \ \ \$ 1. Differentials require a function, $y=f(x)$.
$\ \ \ \$ 2. Differentials require two $x-$values, written as $\ x-$values: $x \ \rightarrow x + \Delta x$, where $x$ is denoted as the "starting" $x-$value and $\Delta x$ can be positive or negative.
$\ \ \ \$ 3. The Exact Change in $y-$values is $\ \Delta y = f(x+ \Delta x)- f(x)$.
$\ \ \ \$ 4. The Differential formula is $\ dy = f'(x) \ \Delta x$, where $x$ is the "starting" $x-$value.
$\ \ \ \$ 5. We will assume that $\ \Delta y \approx dy \$ if $\Delta x$ is "small."

In the list of Differentials Problems which follows, most problems are average and a few are somewhat challenging.

CATEGORY I-- Using Differentials to Estimate the Value of a Numerical Expression

• PROBLEM 1 : Use a Differential to estimate the value of $\ \sqrt{28}$.

• PROBLEM 2 : Use a Differential to estimate the value of $\ 10^{1/3}$.

• PROBLEM 3 : Use a Differential to estimate the value of $\ \sqrt{96}$.

• PROBLEM 4 : Use a Differential to estimate the value of $\ 14^{3/4}$.

• PROBLEM 5 : Use a Differential to estimate the value of $\ \sqrt{72}$.

$\ \ \$ a.) Use 64 as a "starting" $x-$value.
$\ \ \$ b.) Use 81 as a "starting" $x-$value.

• PROBLEM 6 : Use a Differential to estimate the value of $\ e^{-0.3}$.

• PROBLEM 7 : Use a Differential to estimate the value of $\ \ln(1.2)$.

• PROBLEM 8 : Use a Differential to estimate the value of $\ 30^{2/5}$.

• PROBLEM 9 : Use a Differential to estimate the value of $\ \tan( \displaystyle{ \pi \over 4 } + 0.15)$.

• PROBLEM 10 : Use a Differential to estimate the value of $\ \sin( \displaystyle{ \pi \over 6 } - 0.09)$.

• PROBLEM 11 : Use a Differential to estimate the value of $\ \arctan(1.1)$.

• PROBLEM 12 : Use a Differential to estimate the value of $\ \arcsin(0.45)$.

• PROBLEM 13 : Use a Differential to estimate the value of $\ \arcsin(0.12)$.

CATEGORY II-- Using Differentials to Approximate a Relatively Complicated Functional Expression with a Simpler Polynomial Expression

• PROBLEM 14 : Use a Differential to verify the following statement: $\ \ \ \ \sqrt{ 16+3h } \approx 4+ \displaystyle{ 3 \over 8 }h \ \$ for "small" $h$

• PROBLEM 15 : Use a Differential to verify the following statement: $\ \ \ \ \displaystyle{ h^2 \over 4+h^2 } \approx \displaystyle{ 1 \over 4 }h^2 \ \$ for "small" $h$

• PROBLEM 16 : Use a Differential to verify the following statement: $\ \ \ \ (8+5h^3)^{1/3} \approx 2+\displaystyle{ 5 \over 12 }h^3 \ \$ for "small" $h$

• PROBLEM 17 : Use a Differential to verify the following statement: $\ \ \ \ \ln(4+7h) \approx \ln(4)+\displaystyle{ 7 \over 4 }h \ \$ for "small" $h$

• PROBLEM 18 : Use a Differential to verify the following statement: $\ \ \ \ \log(100-h^4) \approx 2 - \displaystyle{ 1 \over 100 \ln(10) }h \ \$ for "small" $h$

• PROBLEM 19 : Use a Differential to verify the following statement: $\ \ \ \ \sqrt{ 25+h^3-h^2 } \approx 5+ \displaystyle{ 1 \over 10 }h^3 - \displaystyle{ 1 \over 10 }h^2 \ \$ for "small" $h$

• PROBLEM 20 : Use a Differential to verify the following statement: $\ \ \ \ \displaystyle{ 1 \over \sqrt{ 1-h^2 } } \approx 1 + \displaystyle{ 1 \over 2 }h^2 \ \$ for "small" $h$

• PROBLEM 21 : Use a Differential to verify the following statement: $\ \ \ \ \displaystyle{ 8-h^2 \over (1+h^2)^2 } \approx 8-17h^2 \ \$ for "small" $h$

CATEGORY II-- Using Differentials to Approximate the Percentage Errors

For the following problems we will refer to $|\Delta x|$ as the "absolute error in $x$ and to $|\Delta y|$ as the "absolute error in $y$. We will define $\displaystyle{ |\Delta x| \over x }$ to be the "absolute percentage error in $x$" and $\displaystyle{ |\Delta y| \over y }$ to be the "absolute percentage error in $y$". For example, if $\Delta x= -0.4$ and $x=20$, then the absolute percentage error in $x$ is $$\displaystyle{ |-0.4| \over 20 }= { 0.4 \over 20 } = { 0.4 \over 20 }{ 5 \over 5} = { 2 \over 100 } = 2\%$$
• PROBLEM 22 : Assume that the edge of a square is measured with an absolute percentage error of at most $3\%$. Use a differential to estimate the absolute percentage error in computing the square's

$\ \ \ \$ a.) $\ \$ perimeter.
$\ \ \ \$ b.) $\ \$ area.

• PROBLEM 23 : Assume that the radius of a circle is measured with an absolute percentage error of at most $2\%$. Use a differential to estimate the absolute percentage error in computing the circle's

$\ \ \ \$ a.) $\ \$ circumference.
$\ \ \ \$ b.) $\ \$ area.

CATEGORY III-- Miscellaneous Differential Problems

• PROBLEM 24 : A large, juicy grapefruit is cut in half. The diameter of the grapefruit (not including the peel) is measured to be 10 inches. The peel is measured to be $1/4$ inch thick.

a.) Find the exact total volume of the peel.
b.) Use a Differential to estimate the volume of the peel.

• PROBLEM 25 : A large angel food cake is in the shape of a cube of side length $30 \ cm$. The cake is dipped in warm chocolate frosting resulting in a delicious coating of frosting $1/2 \ cm$ thick.

a.) Find the exact total volume of the frosting.
b.) Use a Differential to estimate the volume of the frosting.