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Quizzes > Mathematics

Calculus Quiz: Practice Derivatives, Integrals, and More

Quick, 15-question calculus quiz online. Instant answers.

Editorial: Review CompletedCreated By: Lamia LamiaUpdated Aug 24, 2025
Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art showcasing the subject Calculus in a creative and engaging way

This calculus quiz helps you review key ideas and spot gaps before a test. Work through 15 questions on limits, derivatives, integrals, and the Fundamental Theorem; get instant answers and brief tips. For more practice, try our derivative quiz, build basics with a pre calculus quiz, or go deeper with AP calculus unit 1 practice.

What is the limit definition of the derivative of f at x=a using increments h?
lim_{h->0} [f(h)-f(a)]
lim_{x->a} f(x)/x
lim_{x->a} [f(x)-f(a)]/(x-a)
lim_{h->0} [f(a+h)-f(a)]/h
undefined
The derivative of x^n for any real n is n x^{n-1}.
False
True
undefined
If f and g are differentiable, which formula is the product rule?
(fg)' = f'g + g'f' + fg
(fg)' = f'g'
(fg)' = f'g + fg'
(fg)' = f'g - fg'
undefined
The chain rule is used to differentiate compositions of functions.
False
True
undefined
What is d/dx of e^x?
1/e^x
x e^{x-1}
ln(x) e^x
e^x
undefined
d/dx of ln(x) for x>0 equals 1/x.
False
True
undefined
What does the Mean Value Theorem guarantee for a differentiable function on (a,b) continuous on [a,b]?
The function is monotone on [a,b]
The function is convex on [a,b]
There exists c with f''(c)=0
There exists c in (a,b) with f'(c) = [f(b)-f(a)]/(b-a)
undefined
Rolle's Theorem requires f(a)=f(b), continuity on [a,b], and differentiability on (a,b).
True
False
undefined
Which condition identifies a critical point of f?
f'(x)>0
f'(x)=0 or f'(x) does not exist
f(x)=0
f''(x)=0 always
undefined
If f''(x)>0 on an interval, then f is concave up there.
True
False
undefined
What is the second derivative test for a local minimum at a critical point c with f'(c)=0?
If f''(c)>0, f has a local minimum at c
If f''(c)<0, f has a local minimum at c
If f''(c)>0, f has a global minimum at c
If f''(c)=0, f has a local minimum at c
undefined
L'Hospital's Rule applies to 0/0 or infinity/infinity limit forms.
False
True
undefined
For which p does ∫_1^∞ 1/x^p dx converge?
p < 1
all real p
p >= 1
p > 1
undefined
Given y = x^x for x>0, what is dy/dx?
x^x ln x
ln(x)
x^x (ln x + 1)
x^{x-1}
undefined
Which describes Newton's method iteration for solving f(x)=0?
x_{n+1} = f(x_n)
x_{n+1} = x_n - f(x_n)/f'(x_n)
x_{n+1} = x_n - f(x_n)/f'(x_n)
x_{n+1} = x_n - f(x_n)/f'(x_n)
undefined
If f is differentiable and increasing, then f'(x) >= 0 everywhere.
False
True
undefined
What is the arc length of y = f(x) on [a,b]?
∫_a^b sqrt(1 + (f'(x))^2) dx
∫_a^b |f'(x)| dx
∫_a^b (1 + f''(x)) dx
∫_a^b sqrt(1 + f(x)^2) dx
undefined
The surface area of revolution about the x-axis uses 2π y ds with ds = sqrt(1+(y')^2) dx.
False
True
undefined
Which is the curvature κ of a plane curve y=f(x)?
1/|f'(x)|
|f'(x)|/[1+(f''(x))^2]^{1/2}
|f''(x)|/[1+(f'(x))^2]^{3/2}
f''(x)
undefined
The divergence theorem applies to scalar fields.
False
True
undefined
0

Study Outcomes

  1. Apply differentiation techniques to calculate derivatives of various functions.
  2. Utilize integration methods to determine antiderivatives and evaluate Riemann integrals.
  3. Apply the Fundamental Theorem of Calculus to connect derivatives with definite integrals.
  4. Analyze function behavior and sketch curves by identifying key features and critical points.
  5. Evaluate exponential and trigonometric functions within the context of calculus problems.

Calculus Additional Reading

Ready to dive into the world of calculus? Here are some top-notch resources to guide you through the twists and turns of differentiation and integration:

  1. Explore comprehensive lecture notes from MIT's renowned calculus course, covering everything from limits to integrals.
  2. Access a treasure trove of materials, including teaching guides and external links, to enhance your understanding of calculus concepts.
  3. Discover a variety of online courses that delve into the fundamentals of calculus, suitable for learners at all levels.
  4. Peruse detailed lecture notes that provide insights into vectors, partial differentiation, and multiple integrals.
  5. Explore a curated list of free PDFs, cheat sheets, and problem sets to practice and master calculus topics.
Happy studying!
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