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AP®MathsCollege BoardCalculus BCExam QuestionsUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesVector-Valued Functions

Vector-Valued Functions (College Board AP® Calculus BC): Exam Questions

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Vector-Valued Functions

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1
Sme Calculator1 mark

The position vector of a particle moving along a curve in the x y-plane at time t greater or equal than 0 is open angle brackets t to the power of 5 minus 4 t cubed comma space 8 t squared plus 1 close angle brackets.

Find the velocity vector of the particle in terms of time t.

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2
Sme Calculator1 mark

A particle travels along a path in the x y-plane with a velocity vector of open angle brackets e to the power of t minus 4 t comma space e to the power of t plus 4 t close angle brackets, where t is time.

Find the acceleration vector of the particle.

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3
Sme Calculator1 mark

The function f is a vector-valued function defined by f open parentheses t close parentheses equals open angle brackets 1 plus sin space t comma space 3 t squared plus 3 close angle brackets.

Evaluate f apostrophe open parentheses pi over 3 close parentheses.

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4
Sme Calculator1 mark

For time t greater or equal than 0, a particle is traveling in the x y-plane with a velocity vector of open angle brackets ln open parentheses t squared plus 2 close parentheses comma space cos open parentheses e to the power of t close parentheses close angle brackets.

Find the speed of the particle at time t equals 2.4.

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5
Sme Calculator1 mark

For time t greater or equal than 0, a particle moves in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses and velocity vector open angle brackets 3 t squared comma space 1 minus 8 t close angle brackets. A time t equals 0 the particle is at the position open parentheses 2 comma space 5 close parentheses.

Find the position vector of the particle.

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1
Sme Calculator2 marks

A particle moves in the x y-plane with an acceleration vector of open angle brackets 6 e to the power of 2 t end exponent plus sin space t comma space 6 e to the power of negative 2 t end exponent minus sin space t close angle brackets at time t greater or equal than 0. At time t equals 0 the velocity vector is open angle brackets negative 8 comma space 4 close angle brackets.

Find the velocity vector in terms of time, t.

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2
Sme Calculator2 marks

The position vector of a particle moving in the x y-plane at time t greater or equal than 0 is open angle brackets fraction numerator 1 over denominator 3 t plus 1 end fraction comma space 4 t to the power of 3 over 2 end exponent close angle brackets.

Find the speed of the particle at time t equals 0.4.

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3
Sme Calculator3 marks

The position vector of a particle moving along a curve in the x y-plane at time t greater than 1 is open angle brackets sin open parentheses 1 half t close parentheses comma space space t ln space t close angle brackets.

Find the acceleration vector at time t equals pi.

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4
Sme Calculator3 marks

For time t greater or equal than 0, a particle is moving in the x y-plane with a position vector of open angle brackets t cubed comma space cos open parentheses 1 plus t squared close parentheses close angle brackets.

Find the total distance traveled by the particle over the time interval 0 less or equal than t less or equal than 2.5.

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5
Sme Calculator3 marks

At time t greater or equal than 0, a particle moving along a curve in the x y-plane has a position vector of open angle brackets 5 t to the power of 4 minus t comma space 10 e to the power of t cubed end exponent close angle brackets.

For 0 less than t less than 1, there is a point on the curve at which the line tangent to the curve has a slope of 25. Find the time at which the object is at this point.

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1
Sme Calculator3 marks

At time t greater or equal than 0, a particle moves in the x y-plane with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses and an acceleration vector of open angle brackets 20 t cubed plus 8 comma space e to the power of t minus 2 close angle brackets. It is known that, at time t equals 0, the particle is at a position of open parentheses 1 comma space 3 close parentheses with a velocity vector of open angle brackets negative 1 comma space 4 close angle brackets.

Find the position vector of the particle in terms of time, t.

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2a
Sme Calculator4 marks

For time t greater or equal than 0, a particle is moving in the x y-plane along a curve with a position vector of open angle brackets open parentheses 2 t minus 5 close parentheses e to the power of 2 t end exponent comma space 6 t cubed plus 6 close angle brackets.

Find the coordinates of the point on the curve that is the farthest to the left.

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2b
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Explain why there is no point on the curve that is farthest to the right.

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3
Sme Calculator3 marks

A particle moves in the x y-plane with an acceleration vector of open angle brackets 3 t squared cos open parentheses t cubed close parentheses comma space 3 t squared sin open parentheses t cubed close parentheses close angle brackets at time t greater or equal than 0. At time t equals 0, the velocity vector is open angle brackets 1 comma space 0 close angle brackets.

The total distance traveled by the particle in the interval 0 less or equal than t less or equal than 10 is half the total distance traveled by the particle in the interval 10 less than t less or equal than m. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of m.

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4
Sme Calculator3 marks

At time t greater or equal than 0, a particle is moving along a curve in the x y-plane with a position vector of open angle brackets a t plus b plus cos open parentheses 3 t close parentheses comma space p t plus q plus sin open parentheses 3 t close parentheses close angle brackets where a, b, c and d are constants.

Explain why the magnitude of the acceleration never changes throughout the motion of the particle, for t greater or equal than 0.

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5
Sme Calculator3 marks

The velocity vector of a particle moving along a curve in the x y-plane at time t where 0 less or equal than t less than pi over 2 is open angle brackets fraction numerator 1 over denominator 1 plus t squared end fraction comma space fraction numerator e to the power of t over denominator 1 plus e to the power of t end fraction close angle brackets. The position vector of the particle at time t equals 0 is open angle brackets negative 1 comma space ln space 2 close angle brackets.

Show that an equation of the curve followed by the particle is e to the power of y equals 1 plus e to the power of tan open parentheses x plus 1 close parentheses end exponent.

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