38 Calculus questions from the Engineering Mathematics bank, written for Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd.
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A.the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
B.if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
C.the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
D.for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
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2. Which term means: "if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside"?
Junior
A.Cauchy mean value theorem
B.Method of Lagrange multipliers
C.L'Hopital's rule
D.Rolle's theorem
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3. Which statement is correct?
Junior
A.Rolle's theorem — technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
B.Rolle's theorem — expresses a function near a point as a finite polynomial in the displacement plus an explicit remainder term controlling the truncation error
C.Rolle's theorem — if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
D.Rolle's theorem — for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
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4. What is Lagrange mean value theorem?
Junior
A.technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
B.the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
C.for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
D.for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
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5. Which term means: "for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval"?
Junior
A.Lagrange mean value theorem
B.Second derivative test
C.L'Hopital's rule
D.Gamma function
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6. Which statement is correct?
Junior
A.Lagrange mean value theorem — at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
B.Lagrange mean value theorem — for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
C.Lagrange mean value theorem — a point where the second derivative changes sign, so the curve switches between concave up and concave down
D.Lagrange mean value theorem — for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
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7. What is L'Hopital's rule?
Junior
A.at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
B.the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists
C.technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
D.for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
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8. Which term means: "the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists"?
Junior
A.L'Hopital's rule
B.Beta function
C.Gamma function
D.Method of Lagrange multipliers
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9. Which statement is correct?
Junior
A.L'Hopital's rule — the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists
B.L'Hopital's rule — for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
C.L'Hopital's rule — the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
D.L'Hopital's rule — at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
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10. What is Second derivative test?
Junior
A.the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists
B.if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
C.the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
D.at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
Answer + AI explanation with Pro
11. Which term means: "at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive"?
Junior
A.Cauchy mean value theorem
B.Rolle's theorem
C.Lagrange mean value theorem
D.Second derivative test
Answer + AI explanation with Pro
12. Which statement is correct?
Junior
A.Second derivative test — the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists
B.Second derivative test — expresses a function near a point as a finite polynomial in the displacement plus an explicit remainder term controlling the truncation error
C.Second derivative test — the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
D.Second derivative test — at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
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13. What is Point of inflection?
Mid
A.at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
B.a point where the second derivative changes sign, so the curve switches between concave up and concave down
C.for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
D.the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
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14. Which term means: "a point where the second derivative changes sign, so the curve switches between concave up and concave down"?
Mid
A.Point of inflection
B.Taylor's theorem with remainder
C.Euler's theorem on homogeneous functions
D.Gamma function
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15. Which statement is correct?
Mid
A.Point of inflection — a point where the second derivative changes sign, so the curve switches between concave up and concave down
B.Point of inflection — the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
C.Point of inflection — if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
D.Point of inflection — for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
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16. What is Gamma function?
Mid
A.the limit of a quotient in an indeterminate zero-over-zero or infinity-over-infinity form equals the limit of the quotient of the derivatives, when the latter exists
B.expresses a function near a point as a finite polynomial in the displacement plus an explicit remainder term controlling the truncation error
C.the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
D.if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
Answer + AI explanation with Pro
17. Which term means: "the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one"?
Mid
A.Beta function
B.Gamma function
C.Rolle's theorem
D.Lagrange mean value theorem
Answer + AI explanation with Pro
18. Which statement is correct?
Mid
A.Gamma function — the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
B.Gamma function — for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
C.Gamma function — for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
D.Gamma function — the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
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19. What is Euler's theorem on homogeneous functions?
Mid
A.a point where the second derivative changes sign, so the curve switches between concave up and concave down
B.the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
C.the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
D.for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
Answer + AI explanation with Pro
20. Which term means: "for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function"?
Mid
A.Beta function
B.Euler's theorem on homogeneous functions
C.L'Hopital's rule
D.Taylor's theorem with remainder
Answer + AI explanation with Pro
21. Which statement is correct?
Mid
A.Euler's theorem on homogeneous functions — for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
B.Euler's theorem on homogeneous functions — the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
C.Euler's theorem on homogeneous functions — the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
D.Euler's theorem on homogeneous functions — if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
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22. What is Leibniz rule for differentiating an integral?
Mid
A.the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
B.a point where the second derivative changes sign, so the curve switches between concave up and concave down
C.technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
D.for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
Answer + AI explanation with Pro
23. Which term means: "the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits"?
Mid
A.Gamma function
B.Leibniz rule for differentiating an integral
C.Method of Lagrange multipliers
D.Second derivative test
Answer + AI explanation with Pro
24. Which statement is correct?
Mid
A.Leibniz rule for differentiating an integral — the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
B.Leibniz rule for differentiating an integral — if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
C.Leibniz rule for differentiating an integral — for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
D.Leibniz rule for differentiating an integral — expresses a function near a point as a finite polynomial in the displacement plus an explicit remainder term controlling the truncation error
Answer + AI explanation with Pro
25. What is Cauchy mean value theorem?
Mid
A.if a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes somewhere inside
B.for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
C.at a stationary point a negative second derivative gives a local maximum, a positive one a local minimum, and zero is inconclusive
D.for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
Answer + AI explanation with Pro
26. Which term means: "for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point"?
Mid
A.Leibniz rule for differentiating an integral
B.Lagrange mean value theorem
C.Cauchy mean value theorem
D.Rolle's theorem
Answer + AI explanation with Pro
27. Which statement is correct?
Mid
A.Cauchy mean value theorem — the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
B.Cauchy mean value theorem — technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
C.Cauchy mean value theorem — for a function continuous on a closed interval and differentiable inside, some interior point has derivative equal to the average rate of change across the interval
D.Cauchy mean value theorem — for two functions continuous on a closed interval and differentiable inside, the ratio of their total changes equals the ratio of their derivatives at some interior point
Answer + AI explanation with Pro
28. What is Method of Lagrange multipliers?
Senior
A.the definite integral from zero to one of a product of powers of the variable and one minus the variable, expressible as a ratio of gamma functions
B.technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
C.the integral extension of the factorial for which the value at any positive integer n equals the factorial of n minus one
D.the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
Answer + AI explanation with Pro
29. Which term means: "technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients"?
Senior
A.Second derivative test
B.Taylor's theorem with remainder
C.Method of Lagrange multipliers
D.Point of inflection
Answer + AI explanation with Pro
30. Which statement is correct?
Senior
A.Method of Lagrange multipliers — the derivative with respect to a parameter of an integral equals the integral of the partial derivative plus boundary terms from the variable limits
B.Method of Lagrange multipliers — technique for extremising a function subject to constraints by setting the gradient of the function equal to a linear combination of the constraint gradients
C.Method of Lagrange multipliers — for a homogeneous function of degree n, the sum of each variable times its own partial derivative equals n times the function
D.Method of Lagrange multipliers — expresses a function near a point as a finite polynomial in the displacement plus an explicit remainder term controlling the truncation error
Answer + AI explanation with Pro
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