# Applications of differential calculus

## Rates of change

Product Rule: if \$f(x) = uv\$, the \$f'(x) = u'v + uv'\$

## Leibniz's Formula

\$f''(x) = (u'v)' + (uv')'\$

\$ = (u"v + u'v') + (uv" + u'v')'\$

\$ = u"v + 2u'v' + uv"\$

Calculus is used in many fields, not least of which engineering. Mechanical engineering deals with whirly things and that often involves the conversion of force from longitudinal to rotational, or vice-versa. This is known as 'translation', and if it weren't for calculus, the poor engineers would get 'lost in translation'.

The first differential provides a description of momentary rates of change. Graphically, this is the slope of the tangent at a point on a curve. The extreme utility of this tool can best be described by some examples:

### The Ladder and the Wall

This classic problem involves a ladder slipping down a wall. An 8m ladder forms an angle θ with the ground, and is slipping down the wall at the speed of 60cm/s. Determine the rate of change of θ.

We know that \${dy}/{dt} = 0.6m/s\$ and \$y_0 = 8m\$. We need to find an expression for \${dθ}/{dt}\$.

The relationship between the angle and the height \$y\$ is \$sinθ = {y}/{y_0} = y/8\$. By differentiating implicitly we have:

\$\$ cosθ {dθ}/{dt} = 1/8{dy}/{dt}\$\$\$\${dθ}/{dt} = 1/{8cosθ}{dy}/{dt}\$\$

Because \$cosθ = √{1 - sin^2θ} = √{1 - {y^2}/{64}}\$, this becomes:

\$\${dθ}/{dt} = 1/{8√{1 - {y^2}/{64}}}{dy}/{dt}\$\$

### The de L'Hôpital-Bernoulli theorem

Let \$f(x)\$ and \$g(x)\$ be two functions continuous in \$[a;b]\$ and derivable in \$]a;b[\$, and let \$c\$ be in \$]a;b[\$, and \$g(x) ≠ 0\$ for every \$x\$ in \$]a;b[\$ where \$x ≠ c\$.

If \${lim}↙{x→c}f(x) = {lim}↙{x→c}g(x) = 0\$ or \$±∞\$, then \$\${lim}↙{x→c}{f(x)}/{g(x)} = {lim}↙{x→c}{f'(x)}/{g'(x)}\$\$ if \${lim}↙{x→c}{f'(x)}/{g'(x)}\$ exists.

### Economics

Calculus is used in economics for marginal cost analysis. This type of study answers the question of how changing the number of units produced changes the profit and costs of production. Marginal in this sense can be interpreted as 'just one more'.

If increasing the number of units produced results in an increase in profit, it is beneficial to do so. When the production costs and the total revenue are the same, profit is zero. This is the break even point, because investment has been repaid, and a further unit produced would generate net profit.

Marginal profit is the rate of change of profit with respect to the number of units produced or sold.

Marginal revenue is the rate of change of revenue with respect to the number of units sold.

Marginal cost is the rate of change of cost with respect to the number of units sold.

\$r(x)\$ = total revenue from selling \$x\$ amount of units

\$c(x)\$ = total cost of producing \$x\$ amount of units

\$p(x)\$ = profit earned by selling \$x\$ amount of units

\$r'(x)\$ = marginal revenue, the extra revenue for selling one extra unit

\$c'(x)\$ = marginal cost, the extra cost for selling one extra unit

\$p'(x)\$ = marginal profit, the extra profit for selling one more unit

#### Example

The profit for selling \$x\$ units of a product is modeled by: \$p(x)=0.01x^3 + 20x\$ €.

Question: What is the marginal profit for a production of 20 units?

Solution: \$p'(x) = 0.03x^2 + 20\$

\$p'(20) = 0.03(20)^2 + 20 = 32\$ €

An increase from 20 to 21 units of production would result in €32 greater profit.

### Kinematics

The word kinematics comes from Greek (kinema = movement). The distance/displacement, speed/velocity, acceleration and jerk (rate of change of acceleration), of a body moving in a straight line can be mathematically described by simple linear equations.

The velocity of a body can be expressed as a function of its displacement with respect to time: \$v(t) = {ds}/{dt} = s'(t)\$.

The velocity of a body can be expressed as the derivative of its displacement with respect to time: \$v(t) = {ds}/{dt} = s'(t)\$.

The acceleration of a body can be expressed as the second derivative of its displacement, or the first derivative of the velocity, with respect to time: \$a(t) = {dv}/{dt} = {d^2s}/{dt^2} = s″(t)\$.

Associated Mathematicians:

• Isaac Newton
• Gottfried Leibnitz
• Guillaume de l'Hôpital
• Johann Bernoulli (I)

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