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Flashcards: Calculate Rates of Change and Related Rates

A right triangle has sides of length and which are both increasing in length over time such that:

x(t)=2t

y(t)=4t^2

a) Find the rate at which the angle $\theta$ opposite y(t) is changing with respect to time.

$\frac{d\theta}{dt}=\sin(t)+\frac{1}{1+t}$

$\frac{d\theta}{dt}=\frac{2}{1+4t^2}$

$\frac{d\theta}{dt}=\frac{4t}{1-t^2}$

$\frac{d\theta}{dt}=t+\frac{4}{1+2t^2}$

$\frac{d\theta}{dt}=\cos(t)-\frac{1}{1+t}$

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Flashcards: Calculate Rates of Change and Related Rates

A right triangle has sides of length and which are both increasing in length over time such that:

x(t)=2t

y(t)=4t^2

a) Find the rate at which the angle $\theta$ opposite y(t) is changing with respect to time.

Your answer:

$\frac{d\theta}{dt}=\sin(t)+\frac{1}{1+t}$

Your answer:

$\frac{d\theta}{dt}=\frac{4t}{1-t^2}$

Your answer:

$\frac{d\theta}{dt}=t+\frac{4}{1+2t^2}$

Your answer:

$\frac{d\theta}{dt}=\cos(t)-\frac{1}{1+t}$

Correct answer:

$\frac{d\theta}{dt}=\frac{2}{1+4t^2}$

Explanation:

A right triangle has sides of lenght and which are both increasing in length over time such that:

x(t)=2t

y(t)=4t^2

Find the rate at which the angle $\theta$ opposite y(t) is changing with respect to time.

Screen shot 2020 09 04 at 11.29.59 am

a) First, we need to write an expression for the angle $\theta$ as a function of . Because the angle is opposite the side we know that the tangent is simply y/x . Take the inverse of the tangent:

$\theta (t)\, \, =\, \, \tan^{-1}\left(\frac{y(t)}{x(t)} \right )\: \: =\: \: \tan^{-1}\left(\frac{4t^2}{2t}\right)\, \, =\, \, \tan^{-1}\left(2t \right )$

$\theta(t)=\tan^{-1}(2t)$

Now we need to differentiate with respect to .

Recall the general derivative for the inverse tangent function is:

$\frac{d}{dx}\tan^{-1}(x)=\frac{1}{1+x^2}$

Applying this to our function for $\theta(t)$ , and remembering to use the chain rule, we obtain:

$\frac{d\theta}{dt}=\frac{d}{dt}\tan^{-1}(2t)=\frac{2}{1+(2t)^2}=\frac{2}{1+4t^2}$

$\frac{d\theta}{dt}=\frac{2}{1+4t^2}$

Flashcards: Calculate Rates of Change and Related Rates

Soap is sometimes used to determine the location of leaks in industrial pipes. A perfectly spherical soap bubble is growing at a rate of 5 mm^3/sec . What is the rate of change of the surface area of the bubble when the radius of the bubble is 10 mm ?

$80\pi mm^2/sec$

$\frac{1}{80\pi} mm^2/sec$

80 mm^2/sec

1 mm^2/sec

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Flashcards: Calculate Rates of Change and Related Rates

Your answer:

$80\pi mm^2/sec$

Your answer:

$\frac{1}{80\pi} mm^2/sec$

Your answer:

80 mm^2/sec

Correct answer:

1 mm^2/sec

Explanation:

To determine the rate of change of the surface area of the spherical bubble, we must relate it to something we do know the rate of change of - the volume.

The volume of a sphere is given by the following:

$V=\frac{4\pi}{3}r^3$

The rate of change of the volume is given by the derivative with respect to time:

$\frac{\mathrm{d} V}{\mathrm{d} t}=4\pi r^2 \frac{\mathrm{d} r}{\mathrm{d} t}$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x)) \cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

We must now solve for the rate of change of the radius at the specified radius, so that we can later solve for the rate of change of surface area:

$5=4\pi (100) \frac{\mathrm{d} r}{\mathrm{d} t}$

$\frac{\mathrm{d} r}{\mathrm{d} t}=\frac{1}{80\pi}$

Next, we must find the surface area and rate of change of the surface area of the sphere the same way as above:

$A=4\pi r^2$

$\frac{\mathrm{d} A}{\mathrm{d} t}=8 \pi r \frac{\mathrm{d} r}{\mathrm{d} t}$

Plugging in the known rate of change of the surface area at the specified radius, and this radius into the rate of surface area change function, we get

$\frac{\mathrm{d} A}{\mathrm{d} t}=8\pi (10)\cdot \frac{1}{80\pi}= 1mm^2/sec$

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All Calculus AB Resources

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

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All Calculus AB Resources

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

Flashcards: Calculate Rates of Change and Related Rates

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