AP®MathsCollege BoardCalculus ABExam QuestionsUnit 4: Contextual Applications of DifferentiationUnit 4 Overview

Unit 4 Overview (College Board AP® Calculus AB): Exam Questions

36 mins4 questions
1a
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2 marks

Consider the curve defined by the equation x^2 + 3y + 2y^2 = 48. It can be shown that \dfrac{\text{d}y}{\text{d}x} = \dfrac{-2x}{3 + 4y}.

There is a point on the curve near (2, 4) with x-coordinate 3. Use the line tangent to the curve at (2, 4) to approximate the y-coordinate of this point.

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1b
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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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1c
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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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1d
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4 marks

For time t \geq 0, a particle is moving along another curve defined by the equation y^3 + 2xy = 24. At the instant the particle is at the point (4, 2), the y-coordinate of the particle's position is decreasing at a rate of 2 units per second. At that instant, what is the rate of change of the x-coordinate of the particle's position with respect to time?

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2a
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1 mark

Functions f, g, and h are twice-differentiable functions with g\left(2\right) = h\left(2\right) = 4. The line y = 4 + \frac{2}{3}\left(x - 2\right) is tangent to both the graph of g at x = 2 and the graph of h at x = 2.

Find h'\left(2\right).

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2b
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3 marks

Let a be the function given by a\left(x\right) = 3x^{3}h\left(x\right). Write an expression for a'\left(x\right). Find a'\left(2\right).

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2c
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4 marks

The function h satisfies h\left(x\right) = \dfrac{x^{2} - 4}{1 - \left(f\left(x\right)\right)^{3}} for x \neq 2. It is known that \lim_{x \to 2} h\left(x\right) can be evaluated using L'Hospital's Rule. Use \lim_{x \to 2} h\left(x\right) to find f\left(2\right) and f'\left(2\right). Show the work that leads to your answers.

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2d
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1 mark

It is known that g\left(x\right) \leq h\left(x\right) for 1 < x < 3. Let k be a function satisfying g\left(x\right) \leq k\left(x\right) \leq h\left(x\right) for 1 < x < 3. Is k continuous at x = 2? Justify your answer.

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3a
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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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3b
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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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3c
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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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3d
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2 marks

A particle is moving along the curve. At the instant when the particle is at the point \left(\frac{1}{2}, -2\right), its horizontal position is increasing at a rate of \displaystyle \frac{\text{d} x}{\text{d} t} = \frac{2}{3} unit per second. What is the value of \displaystyle \frac{\text{d} y}{\text{d} t}, the rate of change of the particle's vertical position, at that instant?

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4a
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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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4b
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2 marks

There is a point P on the curve G near \left(2, - 1\right) with x-coordinate 1.6. Use the line tangent to the curve at \left(2, - 1\right) to approximate the y-coordinate of point P.

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4c
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2 marks

For x > 0 and y > 0, there is a point S on the curve G at which the line tangent to the curve at that point is vertical. Find the y-coordinate of point S. Show the work that leads to your answer.

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4d
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3 marks

A particle moves along the curve H defined by the equation 2 x y + \text{ln} \; y = 8. At the instant when the particle is at the point \left(4, 1\right), \frac{\text{d}x}{\text{d}t} = 3. Find \frac{\text{d}y}{\text{d}t} at that instant. Show the work that leads to your answer.

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