How to find approximation of rate - CLEP Calculus

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Question

Find the average rate of change for over the interval .

Answer

The rate of change of a function is the amount it changes over a given amount of time.

In mathematical terms, this can be written as

$r=\frac{\Delta y}{\Delta x}$

we plug in our values:

$\frac{5^2+5+1-0^2-0-1}{5-0} = \frac{30}{5}=6$

Note that this is only an average because quadratic functions change at different rates depending on where you are in the function's domain.

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Question

Let $f(x) = ln(e^{x}-sin(\pi -\pi x))$ . Use linear approximation to estimate $f(\frac{3}{2})$ .

Answer

Note that:

Therefore, for values relatively close to 1, we can use the formula for dy (the differential form of the derivative) to estimate f at close values.

$dy= \frac{e^{x}+\pi cos(\pi -\pi x)}{e^{x}-sin(\pi -\pi x)}dx$ From log derivative with chain rule.

Since $\frac{3}{2}$ lies $\frac{1}{2}$ to the right of , $dx = \frac{1}{2}$ and for the estimation, so:

$dy = \frac{e^{3/2}+\pi cos(-\pi )}{e^{3/2}} = 1-\frac{\pi }{e^{3/2}}$

So then:

$f(3/2) \approx1+1-\frac{\pi }{e^{3/2}} = 2-\frac{\pi }{e^{3/2}}$

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Question

Using $(1+x)^{k}\approx1+kx$ , approximate the value of $(0.985)^{50}.$

Answer

First, we need to rearrange the given to match the approximation formula. Therefore, $(0.985)^{50}=(1-0.015)^{50}=1-50(0.015)=0.25.\textup{}$

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Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{7}{30}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{7}{30})^2=\frac{49}{300}$

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Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{13}{15}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{13}{15})^2=\frac{169}{75}$

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Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{1}{30}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{1}{30})^2=\frac{1}{300}$

Compare your answer with the correct one above

Question

Find the average rate of change for over the interval .

Answer

The rate of change of a function is the amount it changes over a given amount of time.

In mathematical terms, this can be written as

$r=\frac{\Delta y}{\Delta x}$

we plug in our values:

$\frac{5^2+5+1-0^2-0-1}{5-0} = \frac{30}{5}=6$

Note that this is only an average because quadratic functions change at different rates depending on where you are in the function's domain.

Compare your answer with the correct one above

Question

Let $f(x) = ln(e^{x}-sin(\pi -\pi x))$ . Use linear approximation to estimate $f(\frac{3}{2})$ .

Answer

Note that:

Therefore, for values relatively close to 1, we can use the formula for dy (the differential form of the derivative) to estimate f at close values.

$dy= \frac{e^{x}+\pi cos(\pi -\pi x)}{e^{x}-sin(\pi -\pi x)}dx$ From log derivative with chain rule.

Since $\frac{3}{2}$ lies $\frac{1}{2}$ to the right of , $dx = \frac{1}{2}$ and for the estimation, so:

$dy = \frac{e^{3/2}+\pi cos(-\pi )}{e^{3/2}} = 1-\frac{\pi }{e^{3/2}}$

So then:

$f(3/2) \approx1+1-\frac{\pi }{e^{3/2}} = 2-\frac{\pi }{e^{3/2}}$

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Question

Using $(1+x)^{k}\approx1+kx$ , approximate the value of $(0.985)^{50}.$

Answer

First, we need to rearrange the given to match the approximation formula. Therefore, $(0.985)^{50}=(1-0.015)^{50}=1-50(0.015)=0.25.\textup{}$

Compare your answer with the correct one above

Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{7}{30}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{7}{30})^2=\frac{49}{300}$

Compare your answer with the correct one above

Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{13}{15}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{13}{15})^2=\frac{169}{75}$

Compare your answer with the correct one above

Question

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{1}{30}$ ?

Answer

Begin by writing the equations for a cube's dimensions. Namely its volume in terms of the length of its sides:

The rates of change of the volume can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{1}{30})^2=\frac{1}{300}$

Compare your answer with the correct one above

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