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Quizzes > Mathematics > Advanced Math

Take the Ultimate Calc Test & Boost Your Calculus Skills

Think you can ace these calculus questions? Dive into basic calculus questions & sample Calculus 1 problems!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration promoting a free calculus quiz on a coral background

Welcome to the Ultimate Calc Test: Conquer Calculus Questions Now! This free calc test invites you to tackle everything from basic calculus questions to challenging sample calculus 1 problems. Along the way, you'll sharpen your grasp of limits, derivatives, integrals and more, discovering where your strengths lie and where fresh practice can help. Ready to ignite your passion for numbers? Jump in and see how many math calculus questions you can master - then push further with our math knowledge quiz or keep the momentum going on tougher calculus 2 practice problems . Let's get started and elevate your calculus confidence today!

Compute the limit of (x^2 - 4)/(x - 2) as x approaches 2.
0
2
4
-4
To evaluate this limit, factor the numerator as (x - 2)(x + 2) and cancel the (x - 2) term. Substituting x = 2 into x + 2 yields 4. This shows the limit exists and equals 4. Learn more about limits
What is the derivative of f(x) = 3x^2?
2x
6x
6
3x^2
Using the power rule, d/dx[x^n] = n x^(n-1). Here n = 2 and the constant factor 3 remains, giving 3 * 2 x^(2-1) = 6x. This is a fundamental result for polynomial differentiation. Read about derivatives
What is the value of the definite integral ?_0^1 x dx?
1/2
2
1
0
The antiderivative of x is x^2/2. Evaluating from 0 to 1 gives (1^2/2) - (0^2/2) = 1/2. This is a standard example in introductory calculus. Integral basics
What is the derivative of a constant function f(x) = C?
x
0
C
Undefined
The derivative measures the rate of change. A constant function does not change, so its rate of change is zero. This is a basic rule in differential calculus. Derivative of constants
What is the value of e^0?
0
e
1
Infinity
By definition of the exponential function, any nonzero base raised to the power zero equals 1. Thus e^0 = 1. This holds for all a^0 where a ? 0. Exponential functions
What is the derivative of sin(x)?
-sin(x)
cos(x)
sin(x)
-cos(x)
The derivative of sin(x) is cos(x), a fundamental result in differential calculus for trigonometric functions. This follows from the limit definition of derivative. Derivative of sine
Evaluate the limit lim_{x?0} (sin(x))/x.
Infinity
Undefined
0
1
This is the standard trigonometric limit; as x approaches 0, sin(x) ~ x, so sin(x)/x ? 1. It is often proved using the unit circle or squeeze theorem. Basic trigonometric limits
Find the derivative of f(x) = x e^x.
e^x + x e^x
x e^(x-1)
x e^x
e^x
Use the product rule: (uv)' = u'v + uv'. Here u = x, v = e^x, so u' = 1 and v' = e^x, giving e^x + x e^x. Product rule
What is the antiderivative of sin(x)?
-cos(x) + C
sin(x) + C
-sin(x) + C
cos(x) + C
Differentiating -cos(x) yields sin(x), so the antiderivative of sin(x) is -cos(x) + C. Constant of integration C is added. Integration of sin
Consider the function f(x) = { sin(x)/x if x ? 0, 1 if x = 0 }. Is f(x) continuous at x = 0?
No
Only from the right
Only from the left
Yes
As x ? 0, sin(x)/x approaches 1, which matches the defined value f(0) = 1. Therefore the function is continuous at x = 0. This is a classic removable discontinuity made continuous by definition. Continuity of functions
What is the derivative of ln(3x)?
1/(3x)
3/x
ln(3) * x
1/x
Use the chain rule: d/dx[ln(u)] = u'/u. Here u = 3x, so u' = 3, giving 3/(3x) = 1/x. Chain rule for logs
Calculate the definite integral ?_1^2 (1/x) dx.
ln(2)
2
ln(1/2)
1
The antiderivative of 1/x is ln|x|. Evaluating from 1 to 2 gives ln(2) - ln(1) = ln(2). Definite integrals
Find lim_{x?0} (1 - cos(x)) / x^2.
1
1/2
0
2
Use the Taylor series for cos(x) or apply L'Hôpital's rule twice. Both yield 1/2 as x ? 0. This limit is common in series expansions. L'Hôpital's rule
What is the derivative of y = x^x?
x^x(ln(x) + 1)
x^x
x^(x-1)
x^x ln(x)
Take ln(y) = x ln(x) and differentiate implicitly: y'/y = ln(x) + 1, so y' = x^x(ln(x)+1). This uses logarithmic differentiation. Derivative of x^x
Evaluate ?_0^? sin^2(x) dx.
1
?/2
?
2
Use the identity sin^2(x) = (1 - cos(2x))/2. Integrate to get [x/2 - sin(2x)/4]_0^? = ?/2. This method simplifies trigonometric integrals. Integrating trigonometric functions
Find the critical points of f(x) = x^3 ? 3x^2 + 2.
x = 1 and x = 2
x = 0 and x = 1
x = 0 and x = 2
x = 2 and x = 3
Compute f'(x) = 3x^2 ? 6x = 3x(x ? 2). Setting f'(x) = 0 gives x = 0 or x = 2. These are the critical points. Critical points
Compute ? x/(x^2 + 1) dx.
arctan(x) + C
(1/2) ln(x^2 + 1) + C
x^2/2 + C
ln(x^2 + 1) + C
Let u = x^2 + 1, then du = 2x dx, so x dx = du/2. The integral becomes (1/2) ? du/u = (1/2) ln|u| + C. U-substitution
What is the derivative of arcsin(x)?
?(1 ? x^2)
x/?(1 ? x^2)
1/(1 ? x^2)
1/?(1 ? x^2)
The derivative of arcsin(x) is 1/?(1 ? x^2) for |x| < 1. This follows from implicit differentiation of sin(y) = x. Derivative of inverse sin
For which values of p does the improper integral ?_1^? 1/x^p dx converge?
p ? 1
p > 1
p = 1
p < 1
The p-integral test shows ?_1^? 1/x^p dx converges only if p > 1. If p ? 1, the area diverges to infinity. This is a key result for improper integrals. Improper integrals
What is the radius of convergence of the power series ?_{n=0}^? x^n / n! ?
?
1
e
0
Apply the ratio test: a_{n+1}/a_n = |x|/(n+1). As n ? ? this goes to 0 for all x, so the series converges for every real x. Thus the radius of convergence is infinite. Radius of convergence
0
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Study Outcomes

  1. Understand Core Calculus Concepts -

    Gain clarity on fundamental ideas like limits, derivatives, and integrals to build a solid foundation for tackling calc test questions.

  2. Apply Derivative Rules -

    Use power, product, quotient, and chain rules to solve a variety of basic calculus questions accurately.

  3. Solve Integration Problems -

    Master common integration techniques, including substitution and integration by parts, for both definite and indefinite integrals.

  4. Analyze Sample Calculus 1 Problems -

    Break down and interpret sample calculus 1 problems to strengthen problem-solving skills and prepare for more advanced questions.

  5. Evaluate Your Strategies with Instant Feedback -

    Use real-time feedback to identify strengths and weaknesses, refining your approach to math calculus questions.

  6. Track and Improve Performance -

    Monitor your score as you progress through the quiz, enabling targeted review and continuous improvement.

Cheat Sheet

  1. Limits & Continuity -

    Mastering limits is essential for your calc test; approach each problem by factoring, rationalizing, or applying the squeeze theorem as shown in MIT OpenCourseWare. Test continuity by checking for jump, removable, or infinite discontinuities, ensuring functions behave predictably at points. A quick mnemonic is "LOTF" (Limit Of Function), reminding you to analyze left and right limits separately.

  2. Derivative Rules & Techniques -

    Familiarize yourself with the power, product, quotient, and chain rules to tackle diverse calculus questions confidently, as detailed in Stewart's Calculus. Practice differentiating polynomial, trigonometric, and composite functions to build speed and accuracy. Remember "dU·V + U·dV" for the product rule to avoid common pitfalls.

  3. Applications of Derivatives -

    Use critical points where f′(x)=0 or undefined to identify local maxima and minima in optimization problems, a staple in math calculus questions from university exams. Employ the first and second derivative tests to determine concavity and inflection points, ensuring a complete curve sketch. Staying systematic with sign charts keeps you organized under time pressure.

  4. Integrals & the Fundamental Theorem of Calculus -

    Link antiderivatives to definite integrals through the Fundamental Theorem of Calculus, converting area computations into algebraic evaluations as outlined by Khan Academy. Practice basic and advanced integration techniques, including substitution and integration by parts, to cover a range of sample calculus 1 problems. Keep the mnemonic "S.U" (Substitute then Undo) to remember u-substitution steps smoothly.

  5. Trig & Exponential Functions -

    Understand derivatives of sin, cos, tan, e^x, and ln(x) to solve both basic calculus questions and more challenging math calculus questions, with reference to official math course guidelines. Memorize key identities like d/dx[e^x]=e^x and d/dx[ln(x)]=1/x, and use triangle diagrams for trig derivatives. Regular timed drills on these functions will boost confidence for any calc test scenario.

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