Graph the damped vibration curve for 0 less than or equals t less than or equals pi .
d left parenthesis t right parenthesis equals e Superscript negative 3 t divided by pi Baseline cosine left parenthesis 2 t right parenthesis
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Part 1
Choose the correct graph where y equals cosine left parenthesis 2 t right parenthesis is black, y equals e Superscript negative 3 t divided by pi is red, y equals negative e Superscript negative 3 t divided by pi is red, and d left parenthesis t right parenthesis equals e Superscript negative 3 t divided by pi Baseline cosine left parenthesis 2 t right parenthesis is blue
A.
-1
1
t
d
piStartFraction pi Over 2 EndFraction
StartFraction pi Over 4 EndFraction
StartFraction 3 pi Over 4 EndFraction
A coordinate system has a horizontal t-axis labeled from 0 to pi in increments of StartFraction pi Over 4 EndFraction and a vertical d-axis labeled from negative 1 to 1 in increments of 1. A smooth red curve falls from left to right at a decreasing rate, passing through (0, 1), (StartFraction pi Over 2 EndFraction, 0.22), and (pi, 0.04). A second smooth red curve rises from left to right at a decreasing rate, passing through (0, negative 1), (StartFraction pi Over 2 EndFraction, negative 0.22), and (pi, negative 0.04). From left to right, a smooth black curve falls from (0, 1) through (StartFraction pi Over 4 EndFraction, 0) to (StartFraction pi Over 2 EndFraction, negative 1) and then rises through (StartFraction 3 pi Over 4 EndFraction, 0) to (pi, 1). A smooth blue curve is bounded by the two red curves, is tangent to the upper red curve at (0, 1) and (pi, 0.04), and is tangent to the lower red curve at (StartFraction pi Over 2 EndFraction, negative 0.22). From left to right, the blue curve falls from (0, 1) through (StartFraction pi Over 4 EndFraction, 0) to a point slightly below and to the left of (StartFraction pi Over 2 EndFraction, negative 0.22), rises through (StartFraction 3 pi Over 4 EndFraction, 0) to a point slightly above and to the left of (pi, 0.04), and then falls through (pi, 0.04). All vertical coordinates are approximate.
B.
-1
1
t
d
piStartFraction pi Over 2 EndFraction
StartFraction pi Over 4 EndFraction
StartFraction 3 pi Over 4 EndFraction
A coordinate system has a horizontal t-axis labeled from 0 to pi in increments of StartFraction pi Over 4 EndFraction and a vertical d-axis labeled from negative 1 to 1 in increments of 1. A smooth red curve falls from left to right at a decreasing rate, passing through (0, 1), (StartFraction pi Over 4 EndFraction, 0.22), and (StartFraction pi Over 2 EndFraction, 0.04). A second smooth red curve rises from left to right at a decreasing rate, passing through (0, negative 1), (StartFraction pi Over 4 EndFraction, negative 0.22), and (StartFraction pi Over 2 EndFraction, negative 0.04). A smooth repeating black curve has a period of StartFraction pi Over 4 EndFraction. From left to right in one period, the black curve falls from (0, 1) through (StartFraction pi Over 8 EndFraction, 0) to (StartFraction pi Over 4 EndFraction, negative 1) and then rises through (StartFraction 3 pi Over 8 EndFraction, 0) to (StartFraction pi Over 2 EndFraction, 1). A smooth blue curve is bounded by the two red curves. The blue curve is tangent to the upper red curve at (0, 1), (StartFraction pi Over 2 EndFraction, 0.22), and (pi, 0.04), and is tangent to the lower red curve at (StartFraction pi Over 4 EndFraction, negative 0.48) and (StartFraction pi Over 2 EndFraction, negative 0.22). From left to right, the blue curve falls from (0, 1) through (StartFraction pi Over 8 EndFraction, 0) to a point slightly below and to the left of (StartFraction pi Over 4 EndFraction, negative 0.48), rises through (StartFraction 3 pi Over 8 EndFraction, 0) to a point slightly above and to the left of (StartFraction pi Over 2 EndFraction, 0.22), falls through (StartFraction 5 pi Over 8 EndFraction, 0) to a point slightly below and to the left of (StartFraction 3 pi Over 4 EndFraction, negative 0.1), rises through (StartFraction 7 pi Over 8 EndFraction, 0) to a point slightly above and to the left of (pi, 0.04), and falls. All vertical coordinates are approximate.
C.
-1
1
t
d
piStartFraction pi Over 2 EndFraction
StartFraction pi Over 4 EndFraction
StartFraction 3 pi Over 4 EndFraction
A coordinate system has a horizontal t-axis labeled from 0 to pi in increments of StartFraction pi Over 4 EndFraction and a vertical d-axis labeled from negative 1 to 1 in increments of 1. A smooth red curve falls from left to right at a decreasing rate, passing through (0, 1), (StartFraction pi Over 2 EndFraction, 0.02), and (pi, 0). A second smooth red curve rises from left to right at a decreasing rate, passing through (0, negative 1), (StartFraction pi Over 2 EndFraction, negative 0.02), and (pi, 0). From left to right, a smooth black curve falls from (0, 1) through (StartFraction pi Over 4 EndFraction, 0) to (StartFraction pi Over 2 EndFraction, negative 1) and then rises through (StartFraction 3 pi Over 4 EndFraction, 0) to (pi, 1). A smooth blue curve is bounded by the two red curves, is tangent to the upper red curve at (0, 1) and (pi, 0), and is tangent to the lower red curve at (StartFraction pi Over 2 EndFraction, negative 0.02). From left to right, the blue curve falls from (0, 1) through (StartFraction pi Over 4 EndFraction, 0) to a point slightly below and to the left of (StartFraction pi Over 2 EndFraction, negative 0.02), rises through (StartFraction 3 pi Over 4 EndFraction, 0) to a point slightly above and to the left of (pi, 0), and then falls through (pi, 0). All vertical coordinates are approximate.
D.
-1
1
t
d
piStartFraction pi Over 2 EndFraction
StartFraction pi Over 4 EndFraction
StartFraction 3 pi Over 4 EndFraction