Sample Calculus Questions
Q1: What is the derivative of f(x) = 5x³ − 3x² + 7?
A. 15x² − 6x ✓
B. 5x² − 3x
C. 15x² − 6x + 7
D. 5x³ − 6x
Q2: Evaluate ∫ 2x dx
A. x² + C ✓
B. 2x² + C
C. x + C
D. 2 + C
Q3: At which x value does f(x) = x² − 4x + 3 have a local minimum?
A. x = 1
B. x = 2 ✓
C. x = 3
D. x = 4
📖 Study Tips for Calculus
1
Build a strong algebra foundation first — most calculus errors are algebra mistakes, not conceptual ones.
2
Memorize the basic derivative rules (power, product, quotient, chain) and practice applying them in the correct order.
3
For integration, always try u-substitution before integration by parts; it handles the majority of standard integrals.
4
Sketch graphs when working with related rates or optimization — a clear diagram prevents misidentifying variables.
❓ Calculus FAQ
What is the Fundamental Theorem of Calculus?
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It links differentiation and integration: Part 1 states that differentiation and integration are inverse operations. Part 2 allows definite integrals to be evaluated using antiderivatives: ∫[a to b] f(x)dx = F(b) − F(a).
What is the difference between a derivative and an integral?
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A derivative measures the instantaneous rate of change of a function (the slope of the tangent line). An integral measures the accumulation of a quantity over an interval (the area under the curve).
How do I know if a function is differentiable at a point?
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A function is differentiable at a point if it is continuous there and the limit of the difference quotient exists from both sides. Sharp corners, cusps, and vertical tangents are not differentiable.
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