Exercises for Question Type 8: Finding the time of change in both directions for a given object

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Exercises for Question Type 8: Finding the time of change in both directions for a given object - IB

Question 1

Find the local maximum of x (t) by determining the time(s) at which x (t) attains an extremum and identifying which is a maximum, given x (t) = 3 + t (9 - t^{2}) .

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Question 2

Parametric equations and speed

Show that the object never comes to rest (i.e. its speed never becomes zero) for: x (t) = 3 + t (9 - t^{2}), y (t) = 4 + t (11 + t)

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Question 3

For the parametric trajectory given by x (t) = 3 + t (9 - t^{2}) y (t) = 4 + t (11 + t) find the times t when the object changes direction in the x -direction.

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Question 4

Determine the interval(s) of t for which the object moves to the right (i.e., \frac{d x}{d t} > 0), given that x (t) = 3 + t (9 - t^{2}) .

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Question 5

Find the time at which the x -coordinate equals 10, given x (t) = 3 + 9 t - t^{3} .

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Question 6

Determine whether there is any time t such that the tangent to the trajectory x (t) = 3 + t (9 - t^{2}), y (t) = 4 + t (11 + t) has slope 1, i.e. \frac{d y}{d x} = 1.

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Question 7

For the trajectory defined by the parametric equations x (t) = 3 + t (9 - t^{2}) y (t) = 4 + t (11 + t) find the time t when the object changes direction in the y -direction.

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Question 8

Solve for the time(s) t when the object crosses the t -axis (i.e., y (t) = 0) for y (t) = t^{2} + 11 t + 4.

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Question 9

Given the function y (t) = 4 + t (11 + t), find the value of t at which y (t) attains its local minimum. Justify your answer.

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Question 10

Determine the interval(s) of t for which the object moves upward (i.e. \frac{d y}{d t} > 0) given y (t) = 4 + t (11 + t) .

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Question 11

A particle moves in the x y -plane such that its position at time t is given by the parametric equations: x (t) = 9 t - t^{3} y (t) = t^{2}

Find the value(s) of t for which the trajectory has a vertical tangent.

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Question 12

Find the time(s) t when the trajectory x (t) = 3 + t (9 - t^{2}), y (t) = 4 + t (11 + t) has a horizontal tangent.

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IB Mathematics Applications & Interpretation Question Type 8: Finding the Time of Change in Both Directions for a Given Object Exercises