• 42 questions

CSET Math 3 Practice Test

Unofficial practice for CSET: Mathematics Subtest III, test code 213: trigonometry, limits, derivatives, integrals and series.

The sine sum formula is applied to 45° + 30°. What exact value does it give for sin 75°?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the California Commission on Teacher Credentialing or Pearson. Bank checked October 9, 2026.

Constructed-response prompts to print 6 lines, made to print

Subtest III extended constructed-response practice

Name ____ Date ____

Write a full answer to each prompt on the lines, showing every step and the reason for it. Unofficial practice prompts; none is from the real test.

Trigonometry, limits and continuity

  • Starting from the Pythagorean theorem on a right triangle inscribed in the unit circle, prove that sin²x + cos²x = 1. Then derive 1 + tan²x = sec²x from it, and say where the second identity is not defined.
  • Use the formal ε–δ definition of a limit to prove that if f(x) approaches L and g(x) approaches M as x approaches a, then f(x) + g(x) approaches L + M.
  • Show that the polynomial p(x) = x³ − 2x + 2 is continuous at x = 1, naming each limit rule you use. Then use the intermediate value theorem to show that p has a zero between −2 and −1.

Derivatives, integrals and series

  • Use the formal definition of the derivative to show that the derivative of sin x is cos x. State the two limits you rely on.
  • Use the limit of right-endpoint Riemann sums to show that the integral of x² from 0 to b is b³/3. Then explain how the fundamental theorem of calculus gives the same value.
  • Derive the formula for the sum of the first n terms of a geometric series with first term a and ratio r, where r is not 1. Then find the condition under which the infinite series converges, and its sum.

Source: Own prompts. The real subtest has 2 extended constructed-response questions on Calculus (https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm); the topics follow the content domains of Subtest III (https://www.ctcexams.nesinc.com/content/docs/CSET_Prep/CS_213subtestdescription.pdf), both read 2026-10-09. Checked October 9, 2026.

How Subtest III is scored 8 lines, made to print

How the written answers count

What the program publishes about scoring, in our words. Unofficial; not a scoring guide of the real test.

The score

  • Two sections, one score The 30 multiple-choice questions carry 70% of the Subtest III score and the 2 extended written answers carry 30%.
  • The scale Raw scores are converted to a 100 to 300 scale, and 220 is the minimum passing score.
  • One sitting A passing score has to be earned at a single administration; sections cannot be combined across sittings.

What scorers look at in a written answer

  • Purpose How far the answer does what the assignment asks.
  • Subject matter knowledge Whether the subject knowledge is accurate and is applied to the task.
  • Support Whether the supporting evidence is fitting and of good quality.
  • Depth and breadth of understanding A fourth characteristic, named for some assignments only.
  • Each written answer is scored by two or more qualified California educators.

Source: Test Results Report Information (https://www.ctcexams.nesinc.com/content/docs/CS_ScoreReport_Insert.pdf) and the test structure (https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm), both read 2026-10-09. Checked October 9, 2026.

Next step More trigonometry: precalculusMore calculus by topic

All 42 questions

Trigonometry 9

  • The angle θ lies in quadrant II and sin θ = 5/13. What is sin 2θ? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • The depth of water at a pier is modeled by d(t) = 5 + 3 cos(πt/6) meters, with t in hours. What are the period of the model and its greatest depth? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • Every term of sin²x + cos²x = 1 is divided by cos²x, where cos x is not 0. Which identity results? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • How many solutions does 2 sin²x − sin x − 1 = 0 have for 0 ≤ x < 2π? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • The inverse sine is applied to sin(5π/6). Which angle does arcsin(sin(5π/6)) give? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • The sine sum formula is applied to 45° + 30°. What exact value does it give for sin 75°? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • What is the exact value of tan(arccos(5/13))? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • Which expression equals cos²x for every real x? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
  • Written in polar form, 1 + i is √2(cos 45° + i sin 45°). By De Moivre's theorem, what is (1 + i)⁸? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0001 Trigonometry (SMR 5.1).
Show all 42 questions (33 more)

Limits and continuity 6

  • A function is defined by f(x) = kx + 1 for x < 2 and f(x) = x² − 1 for x ≥ 2. Which value of k makes f continuous at x = 2? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).
  • As x approaches 2, the quotient (x³ − 8) ÷ (x − 2) approaches which value? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).
  • As x grows without bound, which value does (4x³ + x) ÷ (2x³ − 7x²) approach? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).
  • For f(x) = 4x − 1, the limit as x approaches 2 is 7. In the formal definition with ε = 0.02, what is the largest δ that works? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).
  • The polynomial f(x) = x³ + x − 5 is continuous everywhere. On which interval does the intermediate value theorem guarantee a zero of f? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).
  • What is the limit of sin(5x) ÷ (2x) as x approaches 0? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0002 Limits and Continuity (SMR 5.2).

Derivatives and applications 11

  • A particle moves on a line with position s(t) = t³ − 6t² + 9t meters. What total distance does it travel from t = 0 to t = 3 seconds? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • A spherical balloon is inflated so that its radius grows at 0.5 centimeter per second. How fast is its volume growing when the radius is 10 centimeters? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • For f(x) = √x on the interval from 0 to 4, which number c satisfies the conclusion of the mean value theorem? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • L'Hôpital's rule is used twice on the quotient (eˣ − 1 − x) ÷ x² as x approaches 0. Which limit results? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • The limit of (√(9 + h) − 3) ÷ h as h approaches 0 is the derivative of √x at x = 9. What is its value? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • On which interval is the graph of f(x) = x⁴ − 4x³ concave down? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • One step of Newton's method on f(x) = x³ − 5 starts from x₀ = 2. What is x₁? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • Two nonnegative numbers add up to 12. What is the greatest possible value of the first number times the square of the second? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • What is the derivative of f(x) = x² ln x for x > 0? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • Which function solves dy/dx = 2xy with y(0) = 3? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).
  • Which line is tangent to the curve y = x³ − 4x at the point where x = 2? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0003 Derivatives and Applications (SMR 5.3).

Integrals and applications 9

  • A right-endpoint Riemann sum with four equal subintervals estimates the integral of x² from 0 to 2. What is the sum? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • The limit, as n grows without bound, of the sum from i = 1 to n of (3i/n)² × (3/n) is a definite integral. What is its value? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • Read as area under a graph, what is the integral of |x − 1| from x = 0 to x = 4? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • The region between the line y = x and the parabola y = x² is revolved about the x-axis. What is the volume of the solid? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • To derive the integral of x² from 0 to b from the definition, the Riemann sum is put in closed form. Which formula for 1² + 2² + … + n² is used? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • The upper limit of an integral is a function of x: G(x) is the integral from 1 to x² of √(1 + t³) dt. What is G′(x)? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • What is the area of the region enclosed by the parabola y = 4 − x² and the line y = x + 2? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • What is the integral of x·e^(x²) from x = 0 to x = 1? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).
  • What is the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0004 Integrals and Applications (SMR 5.4).

Sequences and series 7

  • Each series below starts at n = 2. Which one converges? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • In the Maclaurin series of e^(2x), what is the coefficient of x³? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • The ratio test is applied to the series whose nth term is n² ÷ 3ⁿ. What limit does it give, and what follows? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • What is the sum of the arithmetic series 5 + 8 + 11 + … + 302? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • What is the sum of the infinite series 3 − 3/2 + 3/4 − 3/8 + …? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • Which is the Taylor polynomial of degree 2 for ln x centered at x = 1? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).
  • Written as a geometric series, the repeating decimal 0.454545… is 45/100 + 45/10,000 + …. Which fraction is it equal to? Worked twice. Outline: CSET: Mathematics Subtest III content specifications, Calculus 0005 Sequences and Series (SMR 5.5).

How this list was compiled: Our own questions, written to the published content domains of CSET: Mathematics Subtest III: Calculus, test code 213, read 2026-10-09 on the program site of the California Educator Credentialing Assessments (test page https://www.ctcexams.nesinc.com/TestView.aspx?f=HTML_FRAG/CA_CSET211_TestPage.html; test structure https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm; subtest description https://www.ctcexams.nesinc.com/content/docs/CSET_Prep/CS_213subtestdescription.pdf): 30 multiple-choice questions and 2 extended constructed-response questions, all in the one domain Calculus. The structure gives no count for each of the five competencies, so the bank holds 9 on trigonometry, 6 on limits and continuity, 11 on derivatives, 9 on integrals and 7 on sequences and series. Every answer is mathematics, worked a second time. Licence: Questions, choices and explanations are original work of this site.

How it works

Forty-two questions of our own on calculus, the single domain of CSET: Mathematics Subtest III, each with its working shown. This CSET Math 3 practice test is unofficial, with no tie to the California Commission on Teacher Credentialing or Pearson.

Read the full guide

What Subtest III covers

Test code 213 is calculus from start to finish (test structure read October 9, 2026). Exam mode draws 30 of the 42, as many as the real multiple-choice section.

A study guide in five parts

Take the five competencies of the content specifications as a CSET Math Subtest 3 study guide: trigonometry (9 questions here), limits and continuity (6), derivatives and their applications (11), integrals and their applications (9), sequences and series (7).

The written part

The content specifications ask for derivations and proofs, such as the rules of differentiation from the formal definition of the derivative. Nothing here marks writing; six prompts of our own print from under the questions.

Questions and answers

Are these real CSET Math Subtest 3 practice test questions?

No. Every one of the 42 was written for this page from the published calculus competencies. The official sample questions and practice test were left unopened, so nothing here repeats them, and a tally here predicts no real result.

How long is CSET Math Subtest 3?

Two hours for everything. The 30 multiple-choice questions and the 2 extended constructed-response questions share that one session, and the program publishes no separate time for either part.

What score passes CSET Math Subtest 3?

A scaled 220, where the scale runs 100 to 300. Multiple choice is 70% of the subtest score and the two written answers are the other 30%, which is why this page shows a tally and no verdict.

What should a CSET Math Subtest 3 study guide cover?

The five competencies of the calculus domain. Trigonometry is the first: the Pythagorean identities, the sum, double-angle and half-angle formulas, inverse functions, De Moivre's theorem and periodic models. Derivations from formal definitions run through limits, derivatives and integrals.

Is a calculator allowed on CSET Math Subtest 3?

Not a graphing calculator: the test page allows one on Subtest II alone. Some Subtest III assignments do let you handwrite a response on a sheet and scan it in.

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