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CSET Mathematics Subtest III Online Center

This CSET Mathematics Subtest III Online Center brings together practice, lessons, worksheets, flashcards and review tools. Choose the resource that matches the skill you want to work on next.

2 free pop-up tests · 102 flashcards · 5 published competencies
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CSET Mathematics Subtest III Practice Center

Sit one full form of 30 questions with no calculator and a two-hour clock, then read the explanation under every miss. Sort the misses two ways: the ones where you did not know the theorem, and the ones where you knew it and used it on a function that failed its hypotheses. Those need different fixes.

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CSET Mathematics Subtest III at a glance

Subtest III is the calculus subtest, and it is the shortest of the three: 30 multiple-choice questions and 2 constructed-response questions in two hours. One content domain covers everything, from trigonometric identities through Taylor series. Two features shape how you should prepare. There is no calculator, so exact values and clean algebra matter more than they do on Subtest II. And the constructed responses are worth 30 percent of your score, which means you have to write mathematics a reader can follow, not just reach an answer.

Subtest code 213 (Calculus), test version 213-1
Part of CSET: Mathematics, a three-subtest series: Subtest I (211), Subtest II (212), and Subtest III (213). Each subtest is passed independently.
Who needs it Subtest III is required for the full Single Subject Teaching Credential in Mathematics, which asks for Subtests I, II, and III. It is not required for the Foundational-Level Mathematics credential, which asks only for Subtests I and II.
Number of questions 30 multiple-choice questions and 2 constructed-response questions
Content domains One: Calculus. All 30 multiple-choice questions and both constructed responses come from it.
Time limit 2 hours
Calculator None. Calculators are on the CSET prohibited materials list, and the only exception is Subtest II. The vendor does not describe an on-screen calculator for this subtest.
Format Computer-based only, at a test center by appointment, year-round Monday through Saturday. CSET: Mathematics is not on the list of online-proctored tests.
Scoring The multiple-choice section is 70 percent of the subtest score and the two constructed responses are 30 percent. Multiple-choice items are machine scored; constructed responses are read by California educators.
Passing score A scaled score of 220 on a scale of 100 to 300, for each subtest separately.

California publishes a question count for the content domain only, and Subtest III has one domain, so the published count is the whole subtest: 30 multiple-choice and 2 constructed-response questions from Calculus. No percentage or question count is published for the five competencies inside it, so none is claimed here. The practice forms in this Online Center distribute their 30 questions across all five competencies, with derivatives and integrals carrying the most items, which is where the descriptive statements are thickest.

The five competencies, in plain language

California lists five competencies under the Calculus domain, numbered SMR 5.1 through 5.5. Here is what each one asks you to do.

SMR 5.1 Trigonometry

The identities and the unit circle, at the level where you can prove them.

  • Prove that the Pythagorean Theorem is equivalent to sin^2 x + cos^2 x = 1, and that this identity gives 1 + tan^2 x = sec^2 x
  • Prove and apply the sine, cosine, and tangent sum formulas for all real values
  • Analyze properties of trigonometric functions by graphing and by unit circle analysis
  • Apply the definitions and properties of arcsin, arccos, and arctan
  • Apply polar representations of complex numbers, including DeMoivre’s Theorem
  • Model periodic phenomena with periodic functions
  • Recognize equivalent identities, including half-angle and double-angle formulas

SMR 5.2 Limits and continuity

The definition underneath the whole subject, and the theorem that continuity buys you.

  • Derive basic properties of limits and continuity, including the sum, difference, product, constant multiple, and quotient rules, from the formal definition of a limit
  • Show that a polynomial function is continuous at a point
  • Apply the intermediate value theorem, using the geometric meaning of continuity

SMR 5.3 Derivatives and applications

Differentiation from the definition, the three big theorems, and the word problems that use them.

  • Derive the rules of differentiation for polynomial, trigonometric, and logarithmic functions from the formal definition of the derivative
  • Interpret a derivative geometrically, numerically, and analytically
  • Interpret continuous and differentiable functions geometrically and analytically, and apply Rolle’s theorem, the mean value theorem, and L’Hopital’s rule
  • Use the derivative to solve rectilinear motion, related rate, and optimization problems
  • Use the derivative to analyze functions and planar curves, including maxima, minima, and inflection points
  • Solve separable first-order differential equations and apply them to growth and decay

SMR 5.4 Integrals and applications

The definite integral as a limit, the theorem that connects it to antiderivatives, and geometry built from it.

  • Derive definite integrals of standard algebraic functions from the formal definition of the integral
  • Interpret a definite integral geometrically, numerically, and analytically, for example as the limit of Riemann sums
  • Prove the fundamental theorem of calculus and use it to read definite integrals as antiderivatives
  • Apply integrals to compute arc length and the area and volume of geometric figures

SMR 5.5 Sequences and series

Sums you can find exactly, sums you can only test for convergence, and Taylor approximation.

  • Derive and apply the formulas for the sums of finite arithmetic series and of finite and infinite geometric series
  • Determine convergence of a sequence or series using standard techniques such as the ratio, comparison, and integral tests
  • Calculate Taylor series and Taylor polynomials of basic functions

A study routine for a two-hour calculus subtest

Thirty multiple-choice questions in two hours sounds generous until you remember the two constructed responses share that clock and there is no calculator. Budget roughly two and a half minutes per multiple-choice question and leave half an hour for the written work.

DiagnoseTake Practice Test 1 cold: 30 questions, two hours, pencil and paper only. Mark each miss with the competency it came from.
Rebuild the exact valuesWithout a calculator you need sin, cos, and tan of every special angle, the log and exponential facts, and clean radical arithmetic. Run the deck until these come out without hesitation.
Work derivatives and integrals hardestThese two competencies carry the most descriptive statements and the most questions on both practice forms. Related rates and optimization repay drilling more than anything else on the test.
Check hypotheses out loudRolle, the mean value theorem, L’Hopital, and the intermediate value theorem each come with conditions. Say the conditions before you use the theorem, because the wrong answers are built for candidates who skip that step.
Write two full responsesPick one derivative problem and one integral problem and write a complete solution in prose and symbols, the way a reader who cannot see your scratch work would need it. Then compare it to a worked explanation.
Prove itTake Practice Test 2 under the same conditions and confirm the misses are no longer clustered in one competency.

Complete topic map

The first group links to lessons on the site that cover the algebra and function work this subtest leans on constantly. The calculus and trigonometry groups below follow California’s five competencies; there are no linked lessons for those yet, so treat each line as a topic to be able to work cold.

Algebra and function skills you will use throughout

Trigonometry, SMR 5.1 (no lessons on site yet)

  • 15The unit circle and radian measure, with the sine and cosine of every special angle
  • 16Why sin^2 x + cos^2 x = 1 is the Pythagorean Theorem in disguise, and how it gives 1 + tan^2 x = sec^2 x
  • 17Proving the sine, cosine, and tangent sum formulas and applying them to all real values
  • 18Double angle and half angle formulas, and recognizing when two expressions are the same identity
  • 19Graphs of the six trigonometric functions: period, amplitude, midline, and asymptotes
  • 20Inverse trigonometric functions and the restricted ranges that make arcsin, arccos, and arctan single valued
  • 21Modeling periodic phenomena with a sine or cosine function fitted to data
  • 22Polar form of a complex number and DeMoivre’s Theorem for powers and roots

Limits and continuity, SMR 5.2 (no lessons on site yet)

  • 23The formal definition of a limit, and what epsilon and delta each control
  • 24Deriving the sum, difference, product, constant multiple, and quotient rules for limits from that definition
  • 25One-sided limits, infinite limits, and limits as x grows without bound
  • 26Showing that a polynomial function is continuous at a point
  • 27The three ways continuity can fail, and removable against nonremovable discontinuity
  • 28The Intermediate Value Theorem, what it guarantees, and what it does not

Derivatives and applications, SMR 5.3 (no lessons on site yet)

  • 29The derivative as the limit of a difference quotient, and as an instantaneous rate of change
  • 30Deriving the differentiation rules for polynomial, trigonometric, and logarithmic functions from the definition
  • 31Product, quotient, and chain rules, including nested compositions
  • 32Reading a derivative from a graph, from a table of values, and from a formula
  • 33Differentiable implies continuous, plus the corner and cusp examples that show the converse fails
  • 34Implicit differentiation and derivatives of inverse functions
  • 35Rolle’s Theorem and the Mean Value Theorem, with the hypotheses checked before use
  • 36L’Hopital’s rule and the indeterminate forms that let you apply it
  • 37Rectilinear motion: position, velocity, acceleration, and the difference between speed and velocity
  • 38Related rates, from naming the variables to differentiating with respect to time
  • 39Optimization on a closed interval and on an open one, and where the candidates come from
  • 40First and second derivative tests, concavity, and inflection points
  • 41Separable first order differential equations and the growth and decay models they produce

Integrals and applications, SMR 5.4 (no lessons on site yet)

  • 42Riemann sums with left, right, and midpoint rectangles, and the limit that defines the definite integral
  • 43Definite integrals of standard algebraic functions worked from the definition
  • 44Both parts of the Fundamental Theorem of Calculus, including how each is proved
  • 45Antiderivatives, u substitution, and integrals that need algebra before they need calculus
  • 46Area between two curves, integrating in x or in y
  • 47Volume by disks, washers, and shells
  • 48Arc length of a curve

Sequences and series, SMR 5.5 (no lessons on site yet)

  • 49The sum of a finite arithmetic series, derived rather than memorized
  • 50Finite and infinite geometric series, and the condition under which the infinite sum exists
  • 51Convergence of a sequence against convergence of a series, which are different questions
  • 52The ratio test, the comparison test, and the integral test
  • 53The p-series family and the harmonic series, the comparisons you reach for most often
  • 54Taylor polynomials and Taylor series for basic functions, centered where the problem asks

Before test day

Two habits separate a comfortable pass from a near miss. First, do the arithmetic exactly. With no calculator, a candidate who reaches for a decimal approximation halfway through a trigonometry question usually cannot recognize the exact answer among the choices. Second, treat the constructed responses as writing tasks. They are 30 percent of your score, and a reader gives credit for reasoning that is visible on the page: state what you are differentiating, name the theorem you are invoking, and check that its conditions hold. A correct final number with no visible argument earns less than you would expect.

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CSET Mathematics Subtest III: study questions

What can I study in this CSET Mathematics Subtest III Online Center?

This CSET Mathematics Subtest III Online Center includes practice, lessons, worksheets, flashcards and review tools. Topics on the page include SMR 5.1 Trigonometry, SMR 5.2 Limits and continuity, SMR 5.3 Derivatives and applications.

How should I use the CSET Mathematics Subtest III resources?

Choose a topic or practice activity from this Online Center. Review the linked explanation, try the practice, and use the result to decide what to study next. Check each resource’s directions for its format and access requirements.

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