The derivative of a constant function is
- 1
- undefined everywhere
- the constant in every case
- 0
Answer and explanation
0
0
Subjects
Introductory calculus
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Sample exercises
For f(x) = x³, find f′(2).
Answer and explanation
12
f′(x) = 3x²; 3 × 2² = 12.
Which expression is an antiderivative of 2x?
- x² + C
- 2 + C
- x³ + C
- 2x + C
Answer and explanation
x² + C
x² + C
Why include + C in an indefinite integral?
Answer and explanation
Functions differing by an additive constant have the same derivative, so the family of antiderivatives includes an arbitrary constant.
Functions differing by an additive constant have the same derivative, so the family of antiderivatives includes an arbitrary constant.
If velocity is negative on an interval, its signed integral there contributes
- zero in every case
- negative displacement
- positive mass
- necessarily negative distance
Answer and explanation
negative displacement
negative displacement
Does f′(a) = 0 alone prove a local maximum?
- Only when a is positive.
- No.
- Only when the page has a graph.
- Yes, always.
Answer and explanation
No.
No.
Study notes
Power-rule derivatives
Multiply by the exponent and reduce that exponent by one. These tasks use positive integer powers.
Grades 11–12Definite integrals of powers
Use an antiderivative at the upper bound minus its value at the lower bound.
Grades 12–12Limits of polynomials
Polynomials are continuous at every real input, so direct substitution applies.
Grades 11–12A derivative at a point
Differentiate term by term before substituting x.
Grades 11–12Integrate a linear function
Evaluate the antiderivative at the upper limit minus the lower limit.
Grades 11–12Introductory calculus · Formula sheet
Rules, conditions and small examples in one printable reference.
Grades 11–12Calculus: meaning, conditions and checks
A derivative describes local rate of change where it exists.
Grades 11–12Calculus: meaning, conditions and checks · revision card
A derivative describes local rate of change where it exists.
Grades 11–12Introductory calculus: interpret the result
A derivative describes local rate; a definite integral describes signed accumulation.
Grades 11–12Introductory calculus: interpret the result · Revision card
A derivative describes local rate; a definite integral describes signed accumulation.
Grades 11–12Calculus: rules, constants and interpretation
A compact check card for elementary derivatives and definite integrals.
Grades 11–12Derivatives, accumulated change and units
Connect calculations to their meanings and domains.
Grades 11–12Derivatives, accumulated change and units · Revision sheet
Connect calculations to their meanings and domains.
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