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AP Calculus AB : Applications of antidifferentiation

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Example Questions

Example Question #1 : Applications Of Antidifferentiation

Find (dy/dx).

sin(xy) = x + cos(y)

Possible Answers:

None of the above

dy/dx = (1 – cos(xy))/(cos(xy) + sin(y))

dy/dx = (xcos(xy) + sin(y))/(1 – ycos(xy))

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

dy/dx = (cos(xy) + sin(y))/(1 – cos(xy))

Correct answer:

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

Explanation:

The first step of the problem is to differentiate with respect to (dy/dx):

cos(xy)[(x)(dy/dx) + y(1)] = 1 – sin(y)(dy/dx)

*Note: When differentiating cos(xy) remember to use the product rule. (xy' + x'y)

Step 2: Clean the differentiated problem up

cos(xy)(x)(dy/dx) + cos(xy)y = 1 – sin(y)(dy/dx)

cos(xy)(x)(dy/dx) + sin(y)(dy/dx) = 1 – cos(xy)y

Step 3: Solve for (dy/dx)

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

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Example Question #1 : Solving Separable Differential Equations And Using Them In Modeling

Find the equation of the normal line at (2,0) on the graph $y=x^{3}-6x+4$ .

Possible Answers:

y=6x-12

$y'=3x^{2}-6$

y=6x+2

$y=\frac{-1}{6}x+2$

$y=\frac{-1}{6}x+\frac{1}{3}$

Correct answer:

$y=\frac{-1}{6}x+\frac{1}{3}$

Explanation:

The answer is $y=\frac{-1}{6}x+\frac{1}{3}$ .

$y=x^{3}-6x+4$

$y'=3x^{2}-6$

Now plug in x=2 .

$y'=3(2)^{2}-6 = 6$ now we know 6 is the slope for the tangent line. However, we aren't looking for the slope of the tangent line. The slope of the normal line is the negative reciprocal of the tangent's slope; meaning the slope of the normal is $\frac{-1}{6}$ . Now find the equation of the normal line.

$y-0=\frac{-1}{6}(x-2)$

$y=\frac{-1}{6}x+\frac{1}{3}$

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Example Question #2 : Applications Of Antidifferentiation

$f(x) = \frac{x^3}{1-x^2}$

What is the derivative of f(x) ?

Possible Answers:

$\frac{-2x^4-3 x^2(1-x^2)}{(1-x^2)^2}$

$-x$

$\frac{x^2(3-5x^2)}{1-x^2}$

Correct answer:

Explanation:

Use the quotient rule.

${y}' = \frac{(1-x^2)(3x^2)-x^3(-2x)}{(1-x^2)^2} = \frac{x^2(3-3x^2+ 2x^2)}{(1-x^2)^2} = \frac{x^2(3-x^2)}{(1-x^2)^2}$

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Example Question #3 : Solving Separable Differential Equations And Using Them In Modeling

Find $y^{'}$ if

$y=\frac{ln(x)}{x^{3}}$

Possible Answers:

$\frac{3}{x^{3}}$

$y'=\frac{1-3ln(x)}{x^{4}}$

$\frac{1}{x^{4}}$

$\frac{1+3ln(x)}{x^{4}}$

$\frac{ln(x)-1}{x^{4}}$

Correct answer:

$y'=\frac{1-3ln(x)}{x^{4}}$

Explanation:

The answer is

$y'=\frac{1-3ln(x)}{x^{4}}$

$y=\frac{ln(x)}{x^{3}}$

$y'=\frac{(\frac{1}{x})x^{3}-ln(x)(3x^{3})}{x^{6}}$

$y'=\frac{x^{2}(1-3ln(x))}{x^{6}}$

$y'=\frac{1-3ln(x)}{x^{4}}$

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Example Question #3 : Applications Of Antidifferentiation

Find the derivative:

f(x)=x^3-3x^2

Possible Answers:

f'(x)=3x^2-6x

f'(x)=3x^3-6x^2

f'(x)=x^2-6x

f'(x)=3x-6

Correct answer:

f'(x)=3x^2-6x

Explanation:

To find the derivative, multiply the exponent by the coefficent in front of the x term and then decrease the exponent by 1:

f'(x)=3x^2-6x

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Example Question #6 : Solving Separable Differential Equations And Using Them In Modeling

Find the solution to the equation $y'=y$ at $x=2$ with initial condition $y(0)=2$ .

Possible Answers:

$2e^2$

$1$

$e$

$2e$

$e^2$

Correct answer:

$2e^2$

Explanation:

First, we need to solve the differential equation of $y'=y$ .

y'=y

$\frac{dy}{dx}=y$

$\frac{dy}{y}=dx$

$\int \frac{dy}{y}=\int dx$

ln(y)=x+c , where is a constant

y=e^xe^c

y=Ce^x , where is a constant

To find , use the initial condition, y(0)=2 , and solve:

C=2

Therefore, $y=2e^x$ .

Finally, at x=2 , $y(2)=2e^2$ .

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Example Question #7 : Solving Separable Differential Equations And Using Them In Modeling

Solve the differential equation: $\frac{dy}{dx}=-2xy$

Note that (-1,2) is on the curve.

Possible Answers:

y=-2

y=ln(2)+1

y=-x^2+ln(2)+1

$y=2e^{1-x^2}$

Correct answer:

$y=2e^{1-x^2}$

Explanation:

In order to solve differential equations, you must separate the variables first.

$\frac{dy}{y}=-2xdx$

$\int \frac{dy}{y}= \int -2xdx$

ln(y)=-x^2+C

Since point (-1,2) is on the curve, ln(2)=-1+C .

ln(2)+1=C

ln(y)=-x^2+ln(2)+1

To get rid of the log, raise every term to the power of e:

$e^{ln(y)}=e^{-x^2}*e^{ln(2)}*e^1$

$y=2e^{1-x^2}$

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Example Question #6 : Applications Of Antidifferentiation

Suppose $1000 is invested in an account that pays 4.3% interest compounded continuously. Find an expression for the amount in the account after time .

Possible Answers:

$y(t)=1000e^{0.043t}$

$y(t)=1000e^{t}$

$y(t)=4.3e^{1000t}$

$y(t)=4.3e^{t}$

$y(t)=43e^{1000t}$

Correct answer:

$y(t)=1000e^{0.043t}$

Explanation:

The differential equation is $\frac{dy}{dt}=0.043y$ , with boundary condition $y(0)=1000$ .

This is a separable first order differential equation.

$\frac{1}{y}dy=0.043dt$

Integrate both sides.

$ln(y)=0.043t+c$

$e^{ln(y)}=e^{0.043t+c}$

$y=Ce^{0.043t}$

Plug in the initial condition above to see that C=1000 .

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Example Question #8 : Solving Separable Differential Equations And Using Them In Modeling

Find the solution to the differential equation

$\frac{dy}{dx} = \frac{x^{2}+ 4x}{y^{2}}$ when f(0) = 3 .

Possible Answers:

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2}$

$y^{3} = x^{3} + 2x^{2} + 9$

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2} - 27$

$\frac{1}{y} = \frac{1}{x} + 2x^{2} + 9$

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2} + 9$

Correct answer:

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2} + 9$

Explanation:

First, separate the variables of the original differential equation:

$\frac{dy}{dx} = \frac{x^{2}+ 4x}{y^{2}} \Rightarrow y^{2}dy = (x^{2} + 4x)dx$ .

Then, take the antiderivative of both sides, which gives

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2} + C$ .

Use the given condition f(0) = 3 , plugging in

x = 0 and y=3 , to solve for . This gives c=9 , so the correct answer is

$\frac{y^{3}}{3} = \frac{x^{3}}{3} + 2x^{2} + 9$ .

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Example Question #4 : Applications Of Antidifferentiation

Differentiate y=3x^3+5x+7 .

Possible Answers:

y'=9x^2+5x+x

y'=9x^3+5x

y'=6x^2+5

y' = 9x^2+5

Correct answer:

y' = 9x^2+5

Explanation:

While differentiating, multiply the exponent with the coefficient then subtract the exponent by one.

$y'=(3\cdot3)x^{3-1}+(5\cdot1)x^{1-1}+(7\cdot0)$

y'=9x^2+5

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AP Calculus AB : Applications of antidifferentiation

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Applications Of Antidifferentiation

Example Question #1 : Solving Separable Differential Equations And Using Them In Modeling

Example Question #2 : Applications Of Antidifferentiation

Example Question #3 : Solving Separable Differential Equations And Using Them In Modeling

Example Question #3 : Applications Of Antidifferentiation

Example Question #6 : Solving Separable Differential Equations And Using Them In Modeling

Example Question #7 : Solving Separable Differential Equations And Using Them In Modeling

Example Question #6 : Applications Of Antidifferentiation

Example Question #8 : Solving Separable Differential Equations And Using Them In Modeling

Example Question #4 : Applications Of Antidifferentiation

All AP Calculus AB Resources

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