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Wize University Calculus 1 Textbook > Applications of Differentiation

Differentials

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Differentials

Differentials represent infinitesimally small changes in a variable. This concept is directly related to our study of derivatives.

Increments

For a function f(x)f(x), the increment, denoted Δy\Delta y, is the true change in yy when the independent variable varies by Δx=x1−x0\Delta x = x_1 - x_0.

Δy=f(x1)−f(x0)=f(x0+Δx)−f(x0)\boxed{\Delta y = f(x_1) - f(x_0) = f(x_0 +\Delta x) - f(x_0)}

Differentials

For a function f(x)f(x) , dxdx and dydy are called differentials and are related by

dy=f′(x)dx\boxed{dy=f'(x)dx}

Approximations with Differentials

When Δx\Delta x , the change in xx, is small, increments can be approximated by differentials.

Δy≈dy=f′(x)dx\boxed{\Delta y \approx dy = f'(x)dx}


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Example: Differentials

Find an approximation for the percentage change in the area of a circle if its radius is decreased by 2%.

Remember dA≈ΔAdA\approx \Delta A and dr≈Δrdr \approx \Delta r

Percentage Change in r:Δrr=−.02\displaystyle \text{Percentage Change in }r: \\\frac{\Delta r}{r}=-.02


Area of Circle:A=πr2\text{Area of Circle}:\\ A=\pi r^2

  ⟹  dA=2πr∗dr\implies dA=2\pi r*dr

Approximation of change in A:ΔA≈dA=2πrdr≈2πrΔr\displaystyle\text{Approximation of change in }A: \\\Delta A \approx dA=2\pi rdr\approx2\pi r\Delta r


Percentage Change in A:ΔAA≈2πrΔrπr2=2∗Δrr=2(−.02)=−.044% decrease\displaystyle \text{Percentage Change in }A: \\\frac{\Delta A}{A}\approx \frac{2\pi r\Delta r}{\pi r^2}\\ \text{} \\=2*\frac{\Delta r}{r} \\ \text{} \\=2(-.02) \\ \text{} \\=-.04\\ \text{}\\ \boxed{4\% \text{ decrease}}










y=Cxny = Cx^n for constants CC and n≠−1n \neq -1. By approximately what percentage will yy change if xx increases by r%r\%?

Extra Practice
Volume Differential
The volume VV of fluid flowing through a small pipe or tube per unit time at a fixed pressure, is a constant times the fourth power of the tube's radius rr, that is, V=kr4V=kr^4. How will a 10% increase in rr affect VV?
Differentials
Using differentials, estimate the change in the hypotenuse of a right triangle when one of its sides changes from 4 cm to 5 cm and the other changes from 3 cm to 4 cm.
Differentials
If the surface area of a sphere increases by 4%, find the percentage change in the volume.
The volume increases by
6
%.
Differentials
Approximate the percentage change in volume of a sphere if its radius is increased by 3%.
Differentials
The kinetic energy of a moving object is equal to K=12mv2K=\frac{1}{2}mv^2 where m is the mass of the object and v is the speed of the object. Estimate the percentage change in the kinetic energy of the object if its velocity is increased by 0.2%.
Differentials
The volume of a cube increases from 125 cubic centimeters to 130 cubic centimeters.
(a) By approximately how many square centimeters does the surface area increase?
(b) By approximately what percentage does the surface area increase?
Differentials
A balloon is spherical in shape and has radius r. By approximately what percentage will the volume increase if the radius increases by 1.5%?
Practice: Differential
Practice: Differential
Given that y(x)=2x2−x+3y\left(x\right)=2x^2-x+3, find the values of Δx, Δy\Delta x,\ \Delta y and dx, dydx,\ dy from x=0x=0 to x=0.2x=0.2.
Example: Linear functions
For the function y=4x+5y = 4x + 5 the ratio of the change in input over the change in output is a constant. This statement is:
Volume Differential
The volume VV of fluid flowing through a small pipe or tube per unit time at a fixed pressure, is a constant times the fourth power of the tube's radius rr, that is, V=kr4V=kr^4. How will a 10% increase in rr affect VV?
The volume of a cube increases from 125 cubic centimeters to 130 cubic centimeters.
(a) By approximately how many square centimeters does the surface area increase?
(b) By approximately what percentage does the surface area increase?
The kinetic energy of a moving object is equal to K=12mv2K=\frac{1}{2}mv^2 where m is the mass of the object and v is the speed of the object. Estimate the percentage change in the kinetic energy of the object if its velocity is increased by 0.2%.
y=Cxny = Cx^n for constants CC and n≠−1n \neq -1. By approximately what percentage will yy change if xx increases by r%r\%?