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Defining and evaluating functions with different rules on different intervals
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A piecewise function is defined by different formulas on different parts of its domain.
Evaluate the piecewise function at , , and .
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A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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Evaluate at each point:
For : Check which condition: โ (first piece)
For : Check which condition: โ (second piece)
For : Check which condition: โ (third piece)
Answers:
Determine whether the piecewise function is continuous at .
Check continuity at :
Step 1: Find (which piece includes ?) Since includes 1, use second piece:
Find the value of that makes continuous at .
For continuity at , we need:
A function is defined as f(x) = { xยฒ + 1, if x < 0; 2x + 1, if x โฅ 0 }. Evaluate f(-2), f(0), and f(3).
Step 1: Evaluate f(-2): Is -2 < 0? Yes Use f(x) = xยฒ + 1 f(-2) = (-2)ยฒ + 1 = 4 + 1 = 5
Step 2: Evaluate f(0): Is 0 < 0? No Is 0 โฅ 0? Yes Use f(x) = 2x + 1 f(0) = 2(0) + 1 = 1
Step 3: Evaluate f(3): Is 3 < 0? No Is 3 โฅ 0? Yes Use f(x) = 2x + 1 f(3) = 2(3) + 1 = 7
Answer: f(-2) = 5, f(0) = 1, f(3) = 7
For f(x) = { -x + 2, if x โค 1; xยฒ - 2, if x > 1 }, determine if f is continuous at x = 1.
Step 1: Find f(1) using the appropriate piece: Since 1 โค 1, use f(x) = -x + 2 f(1) = -1 + 2 = 1
Step 2: Find the left-hand limit as x approaches 1: lim(xโ1โป) f(x) = lim(xโ1โป) (-x + 2) = -1 + 2 = 1
Step 3: Find the right-hand limit as x approaches 1: lim(xโ1โบ) f(x) = lim(xโ1โบ) (xยฒ - 2) = 1ยฒ - 2 = -1
Step 4: Check continuity conditions: For continuity at x = 1: โข lim(xโ1โป) f(x) = 1 โข lim(xโ1โบ) f(x) = -1 โข f(1) = 1
Step 5: Conclusion: Since lim(xโ1โป) f(x) โ lim(xโ1โบ) f(x), the function is NOT continuous at x = 1. There is a jump discontinuity at x = 1.
Answer: Not continuous (jump discontinuity at x = 1)
Step 2: Find left-hand limit as (use first piece):
Step 3: Find right-hand limit as (use second piece):
Step 4: Compare
Since the left limit () right limit (), the overall limit does not exist.
Answer: The function is NOT continuous at because there is a jump discontinuity (the two pieces don't connect).
Step 1: Find (use second piece since ):
Step 2: Find right-hand limit (use second piece):
Step 3: Find left-hand limit (use first piece with unknown ):
Step 4: Set left limit equal to right limit:
Verify: With :
Answer: makes the function continuous at .