AP®MathsCollege BoardCalculus ABExam QuestionsUnit 3: Differentiation of Composite, Implicit & Inverse FunctionsImplicit Differentiation

Implicit Differentiation (College Board AP® Calculus AB): Exam Questions

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Implicit Differentiation

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1
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2 marks

A cylinder with a diameter of 8 feet contains a liquid. The liquid escapes through a hole in the bottom of the container. The rate of change of the height h of the liquid in the container with respect to time is modelled by fraction numerator d h over denominator d t end fraction equals negative 1 over 25 square root of h, where h is measured in feet and t is measured in minutes.

(The volume of a cylinder with radius r and height h is V equals pi r squared h.)

Find the rate of change of the volume of liquid in the container with respect to time when the height of the liquid is 9 feet. Indicate units of measure.

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2a
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2 marks

Consider the curve given by the equation y squared minus 2 x y equals 5.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator y space over denominator y minus x end fraction.

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2b
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2 marks

Write an equation for the line tangent to the curve at the point left parenthesis 2 comma negative 1 right parenthesis.

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3a
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1 mark

Consider the curve given by the equation x equals sin space y.

Show that fraction numerator d y over denominator d x end fraction equals sec space y.

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3b
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2 marks

Write an equation for the line tangent to the curve at the point open parentheses 0 comma space 0 close parentheses.

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4a
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2 marks

Consider the curve given by the equation tan space 2 y equals x squared y

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x y over denominator 2 sec squared space 2 y minus x squared end fraction.

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4b
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2 marks

Write an equation for the line tangent to the curve at the point open parentheses 0 comma space pi over 2 close parentheses.

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5
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2 marks

Given that y equals arccot space x comma use implicit differentiation to show that fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator x squared plus 1 end fraction.

You may use the results cosec squared space theta equals 1 plus cot squared space theta and fraction numerator d over denominator d x end fraction open parentheses cot space theta close parentheses equals negative cosec squared space theta.

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1a
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2 marks

Consider the curve given by the equation 3 y squared minus 12 equals y space cos space x for y greater than 0.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator y space sin space x over denominator cos space x space minus space 6 y end fraction.

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1b
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1 mark

Write an equation for the line tangent to the curve at the point open parentheses fraction numerator 3 pi over denominator 2 end fraction comma space 2 close parentheses.

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2a
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2 marks

A celestial body changes its temperature, T degrees Fahrenheit, over time, t minutes, according to the relationship:

T squared plus t T equals 500

Find the rate at which the temperature of the body changes, fraction numerator d T over denominator d t end fraction.

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2b
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3 marks

Is the celestial body cooling faster when it is 10 degree straight F or when it is 20 degree straight F ? Explain your reasoning.

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3a
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2 marks

Consider the curve given by the equation 1 fifth x squared e to the power of y equals 5.

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y.

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3b
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2 marks

Find the slope of the tangents to the curve at the two points where the curve intersects the x-axis.

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4
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3 marks

Show that the derivative of y equals arcsin open parentheses 3 x close parentheses is fraction numerator d y over denominator d x end fraction equals fraction numerator 3 over denominator square root of 1 minus 9 x squared end root end fraction.

Show the work that leads to your answer.

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5a
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2 marks

Consider the curve given by the equation ln space y plus x y squared equals 1.

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y.

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5b
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2 marks

The curve defined by ln space y plus x y squared equals 1 passes through the point open parentheses 1 comma space 1 close parentheses.

Write an equation for the line normal to the curve at the point open parentheses 1 comma space 1 close parentheses comma express your final answer in slope-intercept form.

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6a
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2 marks

Is the horizontal line y = 1 tangent to the curve x^2 + 3y + 2y^2 = 48? Give a reason for your answer.

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6b
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1 mark

The curve x^2 + 3y + 2y^2 = 48 intersects the positive x-axis at the point \left(\sqrt{48},\, 0\right). Is the line tangent to the curve at this point vertical? Give a reason for your answer.

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7
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2 marks

Consider the function y = f(x) whose curve is given by the equation 2y^2 - 6 = y\sin x for y > 0.

Show that \dfrac{\text{d}y}{\text{d}x} = \dfrac{y\cos x}{4y - \sin x}.

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1a
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2 marks

Consider the curve given by the equation 8 x y equals 3 plus y to the power of 4.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y over denominator y cubed minus 2 x end fraction.

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1b
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2 marks

A particle is moving along the curve. At the instant when the particle is at the point open parentheses 7 over 2 comma space 3 close parentheses, its horizontal position is increasing at a rate of fraction numerator d x over denominator d t end fraction equals 10 units per second.

What is the value of fraction numerator d y over denominator d t end fraction, the rate of change of the particle's vertical position, at that instant?

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2
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2 marks

Use implicit differentiation to show that the derivative of y equals x to the power of x is fraction numerator d y over denominator d x end fraction equals x to the power of x open parentheses ln space x space plus thin space 1 close parentheses.

Show all steps of your working.

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3
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3 marks

Show that if y equals arccos open parentheses x squared y close parentheses then fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x y over denominator negative x squared minus square root of 1 minus open parentheses x squared y close parentheses squared end root end fraction.

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4a
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3 marks

A treasure chest is being raised from the bottom of the ocean using a pulley system. The depth of the chest below the ocean surface, H comma in meters, and the length of rope unwound from the pulley, R comma in meters, are related by the equation:

H cubed plus t H equals R squared plus 30

where t is the time elapsed, in minutes, since the pulley started operating.

Show that fraction numerator d H over denominator d t end fraction equals fraction numerator 2 R fraction numerator d R over denominator d t end fraction minus H over denominator 3 H squared plus t end fraction and explain what this represents in the context of the problem.

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4b
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3 marks

At t equals 4 minutes, the chest is at a depth of 3meters and the length of the rope is changing at a rate of negative 1 spacemeters per minute.

Find the rate at which the depth of the treasure chest is changing at t equals 4 minutes. Explain whether this means the treasure chest is rising or falling at this time.

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4c
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3 marks

At t equals 5 minutes the length of the rope is now 1 meter and the rate at which the length of rope is changing has not changed. Determine whether the depth of the chest is changing faster or slower than at t equals 4 minutes.

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5a
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2 marks

Consider the curve given by the equation ln open parentheses x y close parentheses plus x y squared equals 1.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator negative y minus x y cubed over denominator x plus 2 x squared y squared end fraction.

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5b
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3 marks

Verify that the point with coordinates open parentheses 1 comma space 1 close parentheses lies on the curve with equation ln open parentheses x y close parentheses plus x y squared equals 1.

Then, find the equation of the tangent to the curve at this point. Express your final answer in slope-intercept form.

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5c
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2 marks

The tangent line to the curve with equation ln open parentheses x y close parentheses plus x y squared equals 1 at the point with coordinates open parentheses 1 comma space 1 close parentheses intercepts the x-axis at the point A and the y-axis at the point B. Find the area of the triangle formed by the origin, O and the points A and B.

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6a
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2 marks

Consider the curve given by the equation 6 x y = 2 + y^3.

Show that \displaystyle \frac{\text{d} y}{\text{d} x} = \frac{2 y}{y^2 - 2 x}.

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6b
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2 marks

Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.

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6c
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3 marks

Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.

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7
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2 marks

Consider the curve G defined by the equation y^{3} - y^{2} - y + \frac{1}{4} x^{2} = 0.

Show that \frac{\text{d}y}{\text{d}x} = \frac{- x}{2 \left(3 y^{2} - 2 y - 1\right)}.

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