Problem 39
Question
Determine the \(t\) intervals on which the curve is concave downward or concave
upward.
$$
x=\sin t, \quad y=\cos t, \quad 0
Step-by-Step Solution
Verified Answer
The curve is concave downward when \( t \) ranges from \( 0 \) to \( \pi \).
1Step 1: Calculate the derivative of \( x \) and \( y \)
First, calculate the derivative of \( x \) and \( y \) w.r.t. \( t \): \( dx/dt = \cos(t) \) and \( dy/dt = -\sin(t) \).
2Step 2: Derive the second derivatives
Next, calculate the second derivative of \( x \) and \( y \): \( d²x/dt² = -\sin(t) \) and \( d²y/dt² = -\cos(t) \).
3Step 3: Calculate the second order derivative of \( y \) w.r.t. \( x \)
Now, calculate the derivative of \( y \) w.r.t. \( x \) through the use of chain rule which gives \( dy/dx = (dy/dt)/(dx/dt) \). Therefore, \( dy/dx = -\sin(t)/\cos(t) = -\tan(t) \). Then, calculate the second order derivative of \( y \) w.r.t. \( x \), \( d²y/dx² = d/dx(dy/dx) \) by applying the chain rule again. This gives \( d²y/dx² = -sec²(t)\). First derivative of \( y \) w.r.t. \( x \), \( dy/dx \), is negative for \( 0
4Step 4: Interpret the result
To summarize: The curve is concave downward for all \( t \) in the given interval \( 0
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