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AP Calculus AB :: 2006-2007 :: Shubleka
Names______________________________ Page 1
Quiz Five – Present your answers neatly for full credit! Problem One (10 points) If the equation of motion of a particle is given by s (t ) A coswt , the particle is said to undergo simple harmonic motion. a) Find the velocity of the particle at time t. b) When is velocity 0? Problem Two
(20 points)
The frequency of vibrations of a vibrating violin string is given by f
1 T , where L is the length of 2L
the string, T is its tension, and is its linear density. a) Find the rate of change of frequency with respect to i) The length (when tension and linear density are constant) ii) The tension (when length and linear density are constant) iii) The linear density (when length and tension are constant) b) The pitch of a note (how high or low the note sounds) is determined by the frequency f . The higher the frequency, the higher the pitch. Use the signs of the derivatives in part a) to determine what happens to the pitch of a note: i) When the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates ii) When the tension is increased by turning a tuning peg iii) When the linear density is increased by switching to another string. Problem Three (10 points) Find an equation of the tangent line to the curve at the given point.
y 4 x x
i)
x 2 y 2 2 x 2 2 y 2 x 2 , 0, 12 This curve is called a cardoid.
ii)
y2
2
2
2
2
5 , 0,2 This is the d evil’s curve.
Problem Four (10 points) How many lines are tangent to both of the parabolas y 1 x 2 and y x 2 1 ? Find all the points at which these tangents touch the parabolas and the equations of the tangents.
The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance. Purcell, E. and Varberg, D.
Names______________________________ Page 1
Quiz Five – Present your answers neatly for full credit! Problem One (10 points) If the equation of motion of a particle is given by s (t ) A coswt , the particle is said to undergo simple harmonic motion. a) Find the velocity of the particle at time t. b) When is velocity 0? Problem Two
(20 points)
The frequency of vibrations of a vibrating violin string is given by f
1 T , where L is the length of 2L
the string, T is its tension, and is its linear density. a) Find the rate of change of frequency with respect to i) The length (when tension and linear density are constant) ii) The tension (when length and linear density are constant) iii) The linear density (when length and tension are constant) b) The pitch of a note (how high or low the note sounds) is determined by the frequency f . The higher the frequency, the higher the pitch. Use the signs of the derivatives in part a) to determine what happens to the pitch of a note: i) When the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates ii) When the tension is increased by turning a tuning peg iii) When the linear density is increased by switching to another string. Problem Three (10 points) Find an equation of the tangent line to the curve at the given point.
y 4 x x
i)
x 2 y 2 2 x 2 2 y 2 x 2 , 0, 12 This curve is called a cardoid.
ii)
y2
2
2
2
2
5 , 0,2 This is the d evil’s curve.
Problem Four (10 points) How many lines are tangent to both of the parabolas y 1 x 2 and y x 2 1 ? Find all the points at which these tangents touch the parabolas and the equations of the tangents.
The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance. Purcell, E. and Varberg, D.
