Functions - CLEP Calculus

Card 0 of 20

Question

Suppose a point on the curve given above has the property that .

Based solely on the graph above, which of the following is most likely the value of the point in question?

Answer

If then the graph must be concave up at the point. Based on the picture, we know that the curve is concave up on at best. The only value that falls on this interval is $-e^{-1}$ , which is $\frac{-1}{e}$ . Since $e\approx2.7$ , this definitely falls on the interval given and we can be sure it is concave up based on the picture.

Compare your answer with the correct one above

Question

What is the critical point for ?

Answer

To find the critical point, you must find the derivative first. To do that, multiply the exponent by the coefficient in front of the and then subtract the exponent by . Therefore, the derivative is: . Then, to find the critical point, set the derivative equal to .

$12x-2=0, x=\frac{1}{6}$ .

Compare your answer with the correct one above

Question

Suppose a point on the curve given above has the property that .

Based solely on the graph above, which of the following is most likely the value of the point in question?

Answer

If then the graph must be concave up at the point. Based on the picture, we know that the curve is concave up on at best. The only value that falls on this interval is $-e^{-1}$ , which is $\frac{-1}{e}$ . Since $e\approx2.7$ , this definitely falls on the interval given and we can be sure it is concave up based on the picture.

Compare your answer with the correct one above

Question

What is the critical point for ?

Answer

To find the critical point, you must find the derivative first. To do that, multiply the exponent by the coefficient in front of the and then subtract the exponent by . Therefore, the derivative is: . Then, to find the critical point, set the derivative equal to .

$12x-2=0, x=\frac{1}{6}$ .

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:

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

Compare your answer with the correct one above

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

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:

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

Compare your answer with the correct one above

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

Question

What is the area of the region bounded by $\small \small y=x^2, x=5$ and $\small y=0$ ?

Answer

To find the area under the curve, we need to perform a definite integral. Essentially, this integral will be summing up all the infinitesimally small rectangles that make up the region. The entire region is in the first quadrant, so we don't have to worry about splitting our region up.

When we take the integral we will need to use,

$\int x^n=\frac{x^{n+1}}{n+1}$ then plug in the upper and lower bounds into the function and take the difference.

Therefore,

$\small \int_{0}^{5}(x^2)dx=(5)^3/3-0^3/3=125/3$

Compare your answer with the correct one above

Question

Find the area of the curve from to $\pi$

Answer

Written in words, solve:

$\int_{0}^{\pi} sin(x) + 2x dx$

To solve:

1. Find the indefinite integral of the function.

2. Plug in the upper and lower limit values and take the difference of the two values.

1. Using the power rule which states,

$\int x^n=\frac{x^{n+1}}{n+1}$ to the term and recalling the integral of is we find,

$\int sin(x) + 2xdx = x^{2} - cos(x)$ .

2. Plug in $\pi$ and for and then take the difference.

$\pi^{2} - cos(\pi) -(0- cos(0) )$ = $\pi^{2} + 2$

note:

$cos(0) = 1 , cos(\pi) = -1$

Compare your answer with the correct one above

Question

What is the average value of the function f(x) = 12x3 + 15x + 5 on the interval \[3, 6\]?

Answer

To find the average value, we must take the integral of f(x) between 3 and 6 and then multiply it by 1/(6 – 3) = 1/3.

The indefinite form of the integral is: 3x4 + 7.5x2 + 5x

The integral from 3 to 6 is therefore: (3(6)4 + 7.5(6)2 + 5(6)) - (3(3)4 + 7.5(3)2 + 5(3)) = (3888 + 270 + 30) – (243 + 22.5 + 15) = 3907.5

The average value is 3907.5/3 = 1302.5

Compare your answer with the correct one above

Question

Find the dot product of a = <2,2,-1> and b = <5,-3,2>.

Answer

To find the dot product, we multiply the individual corresponding components and add.

Here, the dot product is found by:

2 * 5 + 2 * (-3) + (-1) * 2 = 2.

Compare your answer with the correct one above

Tap the card to reveal the answer