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Limits, differentiation and integration
10 ECAT Mathematics questions with answers and explanations.
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What is the limit of sin x / x as x approaches 0?
- 1 0
- 2 1
- 3 Infinity
- 4 It does not exist
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Correct answer: 1
This is a standard limit and its value is 1 (with x in radians).
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What is d/dx of x^3?
- 1 x^2
- 2 3x^2
- 3 3x^3
- 4 x^4 / 4
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Correct answer: 3x^2
By the power rule, the derivative of x^n is n x^(n-1), so the answer is 3x^2.
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What is d/dx of e^(2x)?
- 1 e^(2x)
- 2 2e^(2x)
- 3 2x e^(2x)
- 4 e^(2x) / 2
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Correct answer: 2e^(2x)
By the chain rule, the derivative is e^(2x) multiplied by the derivative of 2x, which is 2.
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What is the integral of x^2 dx?
- 1 x^3 / 3 + C
- 2 2x + C
- 3 x^3 + C
- 4 x^2 / 2 + C
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Correct answer: x^3 / 3 + C
The integral of x^n is x^(n+1) / (n+1) + C, so here it is x^3 / 3 + C.
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What is the value of the definite integral of 3x^2 from 0 to 2?
- 1 6
- 2 8
- 3 12
- 4 4
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Correct answer: 8
The antiderivative is x^3. Evaluating gives 2^3 - 0^3 = 8.
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What is the minimum value of f(x) = x^2 - 4x + 5?
- 1 1
- 2 2
- 3 5
- 4 -1
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Correct answer: 1
Writing it as (x - 2)^2 + 1 shows the minimum is 1, at x = 2.
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What is the limit of (x^2 - 4) / (x - 2) as x approaches 2?
- 1 0
- 2 2
- 3 4
- 4 It does not exist
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Correct answer: 4
Factorise the numerator as (x - 2)(x + 2). Cancelling x - 2 leaves x + 2, which approaches 4.
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What is the slope of the tangent to y = x^2 at x = 3?
- 1 3
- 2 6
- 3 9
- 4 2
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What is the integral of cos x dx?
- 1 -sin x + C
- 2 sin x + C
- 3 -cos x + C
- 4 tan x + C
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Correct answer: sin x + C
The derivative of sin x is cos x, so the integral of cos x is sin x + C.
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What is d/dx of ln x?
- 1 ln x
- 2 x
- 3 1/x
- 4 e^x