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Calculus 1 : How to find rate of change

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Example Question #81 : Rate Of Change

The sides of a rectangle have lengths of 2+t^3 and 1+t^2 . What is the rate of growth of the area at time t=3 ?

Possible Answers:

162

444

145

290

212

Correct answer:

444

Explanation:

The area of the rectangle is given by the product of its sides:

A=wl

A(t)=(2+t^3)(1+t^2)

A(t)=t^5+t^3+2t^2+2

The rate of change of the area can then be found by taking the derivative with respect to time:

We will use the power rule which states,

$x^n\ dx=nx^{n-1}$

to find the following derivative.

$\frac{dA(t)}{dt}=5t^4+3t^2+4t$

$\frac{dA(3)}{dt}=5(81)+3(9)+4(3)$

$\frac{dA(3)}{dt}=405+27+12$

$\frac{dA(3)}{dt}=444$

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Example Question #82 : Rate Of Change

The radius of a circle at any time is given by the equation 3+t^4 . What is the rate of growth of the area of the circle when it has an area of $361\pi$ ?

Possible Answers:

$612\pi$

$608\pi$

$1024\pi$

$512\pi$

$1216\pi$

Correct answer:

$1216\pi$

Explanation:

The area of a circle is given by the equation:

$A=\pi r^2$

$A(t)=\pi(3+t^4)^2$

Find the time that satisfies the area given in the problem:

$361\pi=\pi(3+t^4)^2$

(3+t^4)^2=361

3+t^4=19

t^4=16

t=2

Now that the time is known, to find the rate of area growth, take the derivative of the area equation with respect to time:

We will use the power rule which states,

$x^n\ dx=nx^{n-1}$

and the chain rule,

$f(g(x))dx=f'(g(x))\cdot g'(x)$

to find the following derivative.

$\frac{dA(t)}{dt}=2\pi(4t^3)(3+t^4)$

$\frac{dA(2)}{dt}=2\pi(4(2)^3)(3+(2)^4)$

$\frac{dA(2)}{dt}=2\pi(32)(19)$

$\frac{dA(2)}{dt}=1216\pi$

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Example Question #81 : How To Find Rate Of Change

The two legs of a right triangle are growing at a rate of 2+t^2 and t^3 . What is the rate of growth of the hypotenuse at time t=2 ?

Possible Answers:

$4\sqrt10$

$2\sqrt10$

$2\sqrt5$

Correct answer:

Explanation:

The length of the hypotenuse of a right triangle is given by the Pythagorean Theorem:

c^2=a^2+b^2

c^2(t)=(2+t^2)^2+(t^3)^2

$c(t)=\sqrt{t^6+t^4+4t^2+4}$

Therefore the rate of change of the hypotenuse can be found by taking the derivative of the equation:

We will use the power rule which states,

$x^n\ dx=nx^{n-1}$ and the chain rule,

$f(g(x))dx=f'(g(x))\cdot g'(x)$

to find the following derivative.

$\frac{dc(t)}{dt}=\frac{6t^5+4t^3+8t}{2\sqrt{t^6+t^4+4t^2+4}}$

$\frac{dc(2)}{dt}=\frac{192+32+16}{2\sqrt{64+16+16+4}}$

$\frac{dc(2)}{dt}=\frac{240}{2\sqrt{100}}$

$\frac{dc(2)}{dt}=12$

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Example Question #84 : Rate Of Change

The radius of a sphere at any time is determined by the expression 1+2t^2+t^3 . What is the rate of growth of the sphere's volume at time t=2 ?

Possible Answers:

$6000\pi$

$\frac{19652}{3}\pi$

$23120\pi$

$89632\pi$

$\frac{32000}{3}\pi$

Correct answer:

$23120\pi$

Explanation:

The volume of a sphere is given by the expression:

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

$V(t)=\frac{4}{3}\pi (1+2t^2+t^3)^3$

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

We will use the chain rule,

$f(g(x))dx=f'(g(x))\cdot g'(x)$

to find the following derivative.

$\frac{dV(t)}{dt}=4\pi (4t+3t^2)(1+2t^2+t^3)^2$

$\frac{dV(2)}{dt}=4\pi (20)(17)^2$

$\frac{dV(2)}{dt}=23120\pi$

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Example Question #85 : Rate Of Change

The volume of a cube is growing at a rate of 3+t^2 . What is the rate of growth of the side at time t=3 if the cube has an initial volume of ?

Possible Answers:

$\sqrt[3]{12}$

$\frac{4}{9}$

$\sqrt[3]6$

Correct answer:

$\frac{4}{9}$

Explanation:

We are given the rate of change of the cube's volume:

$\frac{dV(t)}{dt}=3+t^2$

The volume equation can be found by taking the integral of this function with respect to time:

We will use the following rule which states,

$\int x^n\ dx=\frac{x^{n+1}}{n+1}$ to find the following integral.

$V(t)=3t+\frac{1}{3}t^3+C$

The constant of integration can be found by using the initial condition:

$V(0)=3(0)+\frac{1}{3}(0^3)+C=9$

C=9

Giving the volume as a funciton of time:

$V(t)=3t+\frac{1}{3}t^3+9$

Now, find the volume at time t=3 :

$V(3)=3(3)+\frac{1}{3}3^3+9$

V(3)=9+9+9

V(3)=27

From this we can find the length of a side at this time:

V(t)=s(t)^3

s(3)^3=V(3)=27

s(3)=3

Now, let's return the volume equation V(t)=s(t)^3 ; we can relate rates of change by taking the derivative of each side with respect to time:

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

At time t=3

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

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

$\frac{ds(3)}{dt}=\frac{12}{27}$ (Recall that s(3)=3 )

$\frac{ds(3)}{dt}=\frac{4}{9}$

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Example Question #86 : Rate Of Change

Find the rate of change of a line connected by the points (1, 7) and (4, 28) .

Possible Answers:

m=x

m=7

m=0

m=28

m=4

Correct answer:

m=7

Explanation:

To find the rate of change of a line connected by two points. We will use the following equation for slope.

$m=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{28-7}{4-1}=\frac{21}{3}=7$

The rate of change of the line is

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Example Question #87 : Rate Of Change

$z=x^{3}-2y^{2}$ where , , and all change with respect time. When x=3 and y=4 , changes at rate of 5 units/s and changes at rate of 7 units/s . At what rate is changing?

Possible Answers:

$\frac{dz}{dt}=1$

$\frac{dz}{dt}=203$

$\frac{dz}{dt}=23$

$\frac{dz}{dt}=17$

$\frac{dz}{dt}=12$

Correct answer:

$\frac{dz}{dt}=23$

Explanation:

The rate at which changes is $\frac{dz}{dt}$ . The find $\frac{dz}{dt}$ , we will find the first derivative of the equation

$z=x^{3}-2y^{2}$

$\frac{d}{dt}(z)=\frac{d}{dt}\left [x^{3}-2y^{2} \right ]=3x^{2}\frac{dx}{dt}-4y\frac{dy}{dt}$

We are given the following parameters

x=3 , y=4 , $\frac{dx}{dt}=5 units/s$ , and $\frac{dy}{dt}=7 units/s$ .

Subsituting these values into our derivative equation

$\frac{dz}{dt}=3x^{2}\frac{dx}{dt}-4y\frac{dy}{dt}$

$\frac{dz}{dt}=3(3)^{2}(5)-4(4)(7)=135-112=23$

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Example Question #88 : Rate Of Change

A right triangle has two legs of lengths 6in and 8 in . The longer leg is growing at a rate of $0.3\frac{in}{s}$ . What is the rate of change of the shorter leg if the hypotenuse is not currently changing length?

Possible Answers:

$-0.2\frac{in}{s}$

$-0.3\frac{in}{s}$

$-0.5\frac{in}{s}$

$-0.4\frac{in}{s}$

$-0.225\frac{in}{s}$

Correct answer:

$-0.4\frac{in}{s}$

Explanation:

The hypotenuse's length relation to the legs of a right triangle is shown by the Pythagorean Theorem:

a^2+b^2=c^2

This theorem can also be used to show how the rates of change of each parameter relate if we take the derivative of each side of the equation with respect tot ime:

$2a\frac{da}{dt}+2b\frac{db}{dt}=2c\frac{dc}{dt}$ or simply $a\frac{da}{dt}+b\frac{db}{dt}=c\frac{dc}{dt}$

Treating as the smaller side and as the longer side, a= 6in , b=8in , $\frac{db}{dt}=0.3\frac{in}{s}$ . Since the hypotenuse is not changing length at the moment, $\frac{dc}{dt}=0$ . This allows us to plug in terms into the above equation:

$6in\frac{da}{dt}+8in(0.3\frac{in}{s})=c(0)=0$

(In this case, it was unnecessary to calculate that c=10in )

Solving for $\frac{da}{dt}$ :

$6in\frac{da}{dt}=-2.4\frac{in^2}{s}$

$\frac{da}{dt}=-0.4\frac{in}{s}$

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Example Question #89 : Rate Of Change

$z=3x^{2}y^{4}$ where , , and all change with respect time. When x=1 and y=3 , changes at rate of 2 units/s and changes at rate of 4 units/s . At what rate is changing?

Possible Answers:

$\frac{dz}{dt}=57$

$\frac{dz}{dt}=3$

$\frac{dz}{dt}=2268$

$\frac{dz}{dt}=0$

$\frac{dz}{dt}=812$

Correct answer:

$\frac{dz}{dt}=2268$

Explanation:

The rate at which changes is $\frac{dz}{dt}$ . The find $\frac{dz}{dt}$ , we will find the first derivative of the equation

$z=3x^{2}y^{4}$

Since and both change with respect to time, so we must use the product rule when differentiating.

The product rule is $\frac{d}{dt}(ab)=a\frac{d}{dt}(b)+b\frac{d}{dt}(a)$ .

The first derivative of is

$\frac{d}{dt}(z)=\frac{d}{dt}[3x^{2}y^{4}]=3\left [x^{2}\frac{d}{dt}(y^{4})+y^{4}\frac{d}{dt}(x^{2}) \right ]$

$\frac{dz}{dt}=3\left [x^{2}(4y^{3})\frac{dy}{dt}+y^{4}(2x)\frac{dx}{dt} \right ]$

$\frac{dz}{dt}=3\left [4x^{2}y^{3}\frac{dy}{dt}+2xy^{4}\frac{dx}{dt} \right ]$

We are given the following parameters

x=1 , y=3 , $\frac{dx}{dt}=2 units/s$ , and $\frac{dy}{dt}=4 units/s$ .

Subsituting these values into our derivative equation

$\frac{dz}{dt}=3\left [4x^{2}y^{3}\frac{dy}{dt}+2xy^{4}\frac{dx}{dt} \right ]$

$\frac{dz}{dt}=3\left [4(1)^{2}(3)^{3}(4)+2(1)(3)^{4}(2) \right ]$

$\frac{dz}{dt}=3\left [432+324 \right ]=2268$

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Example Question #90 : Rate Of Change

Find the rate of change of a line connected by the points (0, 4) and (12, -7) .

Possible Answers:

$\frac{-11}{12}$

$\frac{-12}{7}$

$\frac{1}{12}$

Correct answer:

$\frac{-11}{12}$

Explanation:

To find the rate of change of a line connected by two points. We will use the following equation for slope.

$m=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-7-4}{12-0}=\frac{-11}{12}$

The rate of change of the line is $\frac{-11}{12}$ .

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Calculus 1 : How to find rate of change

Study concepts, example questions & explanations for Calculus 1

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Example Question #81 : Rate Of Change

Example Question #82 : Rate Of Change

Example Question #81 : How To Find Rate Of Change

Example Question #84 : Rate Of Change

Example Question #85 : Rate Of Change

Example Question #86 : Rate Of Change

Example Question #87 : Rate Of Change

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