Piecewise continuity and the Intermediate Value Theorem
Grade 12 Math Worksheet - Challenge Me
Circle the best answer, or write your answer on the line.
f(x) = x² for x < 1, ax + b for 1 ≤ x < 3, and x² for x ≥ 3. Ari makes f continuous. As (a, b), what are a and b?
f(x) = x² for x < 2, ax + b for 2 ≤ x < 4, and x² for x ≥ 4. Bo makes f continuous. As (a, b), what are a and b?
f(x) = x² for x < -1, ax + b for -1 ≤ x < 2, and x² for x ≥ 2. Esme makes f continuous. As (a, b), what are a and b?
f(x) = x² for x < 0, ax + b for 0 ≤ x < 3, and x² for x ≥ 3. Jun makes f continuous. As (a, b), what are a and b?
f(0) = 6; f(1) = -4; f(2) = 7; f(3) = -3; f(4) = 9. Kira studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?
- at least 4 zeros
- at least 2 zeros
- exactly 4 zeros
f(0) = -8; f(1) = 3; f(2) = -5; f(3) = 7; f(4) = -9. Noah studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?
- at least 3 zeros
- exactly 4 zeros
- at least 4 zeros
The air is 12 °C at 6 a.m. and 25 °C at noon, changing continuously. Pia asks whether it is ever exactly 20 °C. Which is true?
- not promised: 20 °C is outside the range
- yes, at some moment it equals 20 °C
- yes, if it changes at a steady rate
A drone climbs continuously from 20 m to 80 m in 10 seconds. Ren asks whether it is ever exactly 95 m. Which is true?
- yes, at some moment it equals 95 m
- not promised: 95 m is outside the range
- yes, if it changes fast enough
Soup cools continuously from 90 °C to 40 °C in an hour. Tala asks whether it is ever exactly 60 °C. Which is true?
- yes, if it changes at a steady rate
- yes, at some moment it equals 60 °C
- not promised: 60 °C is outside the range
A pool's depth drains continuously from 5 ft to 2 ft in a day. Zee asks whether it is ever exactly 1 ft. Which is true?
- yes, if it changes fast enough
- not promised: 1 ft is outside the range
- yes, at some moment it equals 1 ft