Grade 12 · Math · Calculus

12th grade Piecewise continuity and the Intermediate Value Theorem worksheet

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

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Piecewise continuity and the Intermediate Value Theorem

Grade 12 Math Worksheet

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f(x) = kx + 1 for x < 2 and f(x) = x² + 5 for x ≥ 2. Ari picks k so f is continuous at x = 2. What is k?
f(x) = kx − 2 for x < 3 and f(x) = x² + 1 for x ≥ 3. Bo picks k so f is continuous at x = 3. What is k?
f(x) = kx + 4 for x < 3 and f(x) = x² + 6 for x ≥ 3. Esme picks k so f is continuous at x = 3. What is k?
A ramp is h(x) = 0.5x m high for x < 4, then a flat deck at height c for x ≥ 4. Kira wants no step. What c makes it continuous, in m?
A cart is at s(t) = 12t m for t < 5 s, then stops at s = c for t ≥ 5. Milo wants no step. What c makes it continuous, in m?
A slide drops d(x) = 0.25x m for x < 8, then runs flat at depth c for x ≥ 8. Noah wants no step. What c makes it continuous, in m?
7.
f(0) = 4; f(1) = 1; f(2) = -2; f(3) = -5; f(4) = -3. Pia uses the table shown and the IVT. Where must f have a zero?
- between 1 and 2
- between 0 and 1
- between 2 and 3
8.
f(0) = -3; f(1) = -2; f(2) = -1; f(3) = 4; f(4) = 6. Ren uses the table shown and the IVT. Where must f have a zero?
- between 3 and 4
- between 2 and 3
- between 0 and 1
9.
f(0) = 7; f(1) = 5; f(2) = 2; f(3) = 1; f(4) = -2. Tala uses the table shown and the IVT. Where must f have a zero?
- between 3 and 4
- between 1 and 2
- between 2 and 3
10.
f(0) = -6; f(1) = -4; f(2) = -1; f(3) = 3; f(4) = 4. Zee uses the table shown and the IVT. Where must f have a zero?
- between 1 and 2
- between 2 and 3
- between 0 and 1

Piecewise continuity and the Intermediate Value Theorem. Every version has 10 questions, and the answer key prints on the second page.

I can check a piecewise function at its join and find f at the join.
I can choose constants that make a piecewise function continuous.
I can use the IVT to promise a zero or a value, and say when it promises nothing.

Standard: none (beyond CCSS)

join: The x where a piecewise function changes rule.

Intermediate Value Theorem (IVT): A continuous f on [a, b] hits every height between f(a) and f(b).

sign change: Neighboring values on opposite sides of 0.

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Thought the join point is missing. One of the pieces always includes it (≥ or ≤).

Checked only one piece at the join. Both pieces must give the same height.

What is the most common mistake with piecewise continuity and the intermediate value theorem?

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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