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Mathematics · EUEE topic

Calculus: every EUEE question 2015–2017 E.C. (31 questions), solved

31 questions · 2015 ×19 · 2016 ×5 · 2017 ×7

Limits, derivatives, extrema, and rates of change from every EUEE Mathematics paper in the Abrari bank.

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2015 E.C.

2015 E.C. · natural · Q36 · Limits

What is the value of lim⁡x→1x2−1x2+1\lim_{x \to 1} \frac{x^{2} - 1}{x^{2} + 1}?
  1. A. 3
  2. B. 1
  3. C. 2
  4. D. 0
Worked solutionUnofficial

Answer: D

The rational function x2−1x2+1\frac{x^{2}-1}{x^{2}+1} is defined at x=1x=1 because the denominator is 12+1=2≠01^{2}+1=2\neq 0. The limit is ordinary substitution.
lim⁡x→1x2−1x2+1=12−112+1=02=0.\lim_{x\to 1}\frac{x^{2}-1}{x^{2}+1}=\frac{1^{2}-1}{1^{2}+1}=\frac{0}{2}=0.
Letter D is 00.

Why not A/B/C?

  • A. 33 comes from adding the numerator and the denominator at x=1x=1 (or adding 1+21+2) instead of dividing 00 by 22.
  • B. 11 would be x2−1x2−1\frac{x^{2}-1}{x^{2}-1} or a mistaken 2/22/2. The denominator is x2+1x^{2}+1, so the value is 0/2=00/2=0.
  • C. 22 is the value of the denominator at x=1x=1, not the value of the fraction.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q42 · Limits of sequences

Let the sequence {an} converges to 3 while {bn} converges to -1. Then the sequence {an-bn} converges to
  1. A. 1
  2. B. 4
  3. C. 3
  4. D. 1/3
Worked solutionUnofficial

Answer: B

Answer B: 4. The stem is: Let the sequence {an} converges to 3 while {bn} converges to -1. Then the sequence {an-bn} converges to Letter B is the Abrari key for this Limits of sequences.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q45 · Integration

Which of the following is equal to ∫3cos⁡3x dx∫3\cos 3x\,dx?
  1. A. 13sin⁡3x+c\frac{1}{3}\sin 3x + c
  2. B. sin⁡3x+c\sin 3x+c
  3. C. sin⁡x+c\sin x+c
  4. D. −sin⁡x+c-\sin x+c
Worked solutionUnofficial

Answer: B

Answer B: sin⁡3x+c\sin 3x+c. The stem is: Which of the following is equal to ∫3cos⁡3x dx∫3\cos 3x\,dx? Letter B is the Abrari key for this Integration.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q47 · Fundamental theorem of calculus

Let f be a continuous function on [0, 1] and let F be an anti-derivative of f with F(1)=−1F(1) = -1 and F(0)=1F(0) = 1. Then ∫01f(x) dx\int_{0}^{1} f(x) \, dx is equal to
  1. A. -22
  2. B. 11
  3. C. 0
  4. D. 22
Worked solutionUnofficial

No keyed letter is published for this item. No keyed letter is published for this 2015 E.C. natural item (Fundamental theorem of calculus). F(1)-F(0) = -1-1 = -2; no option matches (-2). Options appear corrupted.Try the options yourself, then practise the same subject timed in the app.

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2015 E.C. · natural · Q59 · Antiderivatives

Let F be the anti-derivative of f. Then which of the following is NOT true about F and f?
  1. A. ddx∫F(x) dx=f(x)\frac{d}{dx} \int F(x) \, dx = f(x)
  2. B. F′(x)=f(x)F^{'}(x) = f(x)
  3. C. ∫F(x) dx=f(x)+c\int F(x) \, dx = f(x) + c
  4. D. ∫ddxf(x) dx=f(x)+c\int \frac{d}{dx} f(x) \, dx = f(x) + c
Worked solutionUnofficial

Answer: C

Answer C: ∫F(x) dx=f(x)+c\int F(x) \, dx = f(x) + c. The stem is: Let F be the anti-derivative of f. Then which of the following is NOT true about F and f? Letter C is the Abrari key for this Antiderivatives.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q60 · Derivatives (tangent slopes)

Let f(x)=1xf(x) = \frac{1}{\sqrt{x}} and g(x)=2−16x3g(x) = 2 - \frac{1}{6}x^{3} be two functions. For what values of x do the graphs of f and g have tangent lines of the same slope?
  1. A. 32\frac{\sqrt{3}}{2}
  2. B. 1
  3. C. 4
  4. D. 1/4
Worked solutionUnofficial

Answer: B

Answer B: 1. The stem is: Let f(x)=1xf(x) = \frac{1}{\sqrt{x}} and g(x)=2−16x3g(x) = 2 - \frac{1}{6}x^{3} be two functions. For what values of x do the graphs of f and g have tangent lines of the same slope? Letter B is the Abrari key for this Derivatives (tangent slopes).

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q62 · Derivatives

Let f(x)=3x+2x−sin⁡x+exf(x) = 3x + 2x - \sin{x} + e^{x}. What is the value of f′(0)f^{'}(0)?
  1. A. 2
  2. B. 1
  3. C. -1
  4. D. 4
Worked solutionUnofficial

No keyed letter is published for this item. No keyed letter is published for this 2015 E.C. natural item (Derivatives). As written f'(0)=5-cos0+e^0=5; no option matches. Function text likely garbled (3x+2x).Try the options yourself, then practise the same subject timed in the app.

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2015 E.C. · natural · Q63 · Chain rule

Let f and g be differentiable functions, and h(x)=g(f(3x+1))h(x) = g(f(3x+1)). If f′(1)=f(1)=2f^{'}(1) = f(1) = 2 and g′(2)=1g^{'}(2) = 1, then what is the value of h′(0)h^{'}(0)?
  1. A. 2
  2. B. 1
  3. C. 6
  4. D. 4
Worked solutionUnofficial

Answer: C

Answer C: 6. The stem is: Let f and g be differentiable functions, and h(x)=g(f(3x+1))h(x) = g(f(3x+1)). If f′(1)=f(1)=2f^{'}(1) = f(1) = 2 and g′(2)=1g^{'}(2) = 1, then what is the value of h′(0)h^{'}(0)? Letter C is the Abrari key for this Chain rule.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · natural · Q64 · Concept of derivative

Which of the following describes the mechanical meaning of derivative?
  1. A. Gradient of graphs of functions
  2. B. Area under the curve
  3. C. Instantaneous rate of change
  4. D. Slope of tangent line
Worked solutionUnofficial

Answer: C

Answer C: Instantaneous rate of change. The stem is: Which of the following describes the mechanical meaning of derivative? Letter C is the Abrari key for this Concept of derivative.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q47 · Limits

Which one of the following limit values is correct for the limit of the given function?
  1. A. lim⁡x→2x3−82−x=12\lim_{x\to 2}\frac{x^3 - 8}{2 - x} = 12
  2. B. lim⁡x→11−x2x−1=−2\lim_{x\to 1}\frac{1-x^2}{x-1} = -2
  3. C. lim⁡x→102x=100\lim_{x\to 10}2^x = 100
  4. D. lim⁡x→5x2−5x4−25=15\lim_{x\to \sqrt{5}}\frac{x^2 - 5}{x^4 - 25} = \frac{1}{5}
Worked solutionUnofficial

Answer: B

Answer B: lim⁡x→11−x2x−1=−2\lim_{x\to 1}\frac{1-x^2}{x-1} = -2. The stem is: Which one of the following limit values is correct for the limit of the given function? Letter B is the Abrari key for this Limits. (1-x^2)/(x-1) = -(x+1) → -2 at x=1. A actually equals -12; C=1024; D=1/10.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q48 · Limits

If ff, gg, hh, and mm are functions defined by f(x)=x+7x2+9x+14f(x)=\frac{x+7}{x^2+9x+14}, g(x)=cos⁡xxg(x)=\frac{\cos x}{\sqrt{x}}, h(x)=5x−1x−3h(x)=\sqrt{\frac{5x-1}{x-3}}, and m(x)=sin⁡xcos⁡xxm(x)=\frac{\sin x \cos x}{x}, which one of the following is the correct result?
  1. A. lim⁡x→0m(x)=0\lim_{x\to 0}m(x) = 0
  2. B. lim⁡x→−7f(x)=−15\lim_{x\to -7}f(x) = -\frac{1}{5}
  3. C. lim⁡x→3+h(x)=15\lim_{x\to 3^+}h(x) = \sqrt{15}
  4. D. lim⁡x→0+g(x)=1\lim_{x\to 0^+}g(x) = 1
Worked solutionUnofficial

Answer: B

Answer B: lim⁡x→−7f(x)=−15\lim_{x\to -7}f(x) = -\frac{1}{5}. The stem is: If ff, gg, hh, and mm are functions defined by f(x)=x+7x2+9x+14f(x)=\frac{x+7}{x^2+9x+14}, g(x)=cos⁡xxg(x)=\frac{\cos x}{\sqrt{x}}, h(x)=5x−1x−3h(x)=\sqrt{\frac{5x-1}{x-3}}, and m(x)=sin⁡xcos⁡xxm(x)=\frac{\sin x \cos x}{x}, which one of the following is the correct result? Letter B is the Abrari key for this Limits. f(x) = 1/(x+2) after cancellation; limit at x=-7 is 1/(-5) = -1/5. A equals 1; C diverges to +∞; D diverges to +∞.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q50 · Derivatives - concept

Which one of the following is NOT true about the derivative of a function?
  1. A. The derivative of a function at a point is equal to the area of the region enclosed by the graph of the function and the coordinate axes.
  2. B. The derivative of a function ff at a point x0x_0 is the instantaneous rate of change of f(x)f(x) at (x0,f(x0))(x_0, f(x_0)).
  3. C. The derivative of a function gg at a point tt is equal to the limit of the ratio of the difference of functional values of gg at xx and at tt to the difference x−tx - t as xx approaches tt.
  4. D. The derivative of a function ff at a point xx is the slope of a line tangent to the graph of the function at the point (x,f(x))(x, f(x)).
Worked solutionUnofficial

Answer: A

Answer A: The derivative of a function at a point is equal to the area of the region enclosed by the graph of the function and the coordinate axes.. The stem is: Which one of the following is NOT true about the derivative of a function? Letter A is the Abrari key for this Derivatives - concept.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q51 · Average rate of change

The following table is the distance traveled by a car in the given times. Which of the following time intervals has the maximum average rate of change of the distance with respect to time?

This item uses a figure in the original paper (not redrawn here).

  1. A. t = 1 hour and t = 2 hours
  2. B. t = 2 hours and t = 3 hours
  3. C. t = 3 hours and t = 4 hours
  4. D. t = 4 hours and t = 5 hours
Worked solutionUnofficial

No keyed letter is published for this item. No keyed letter is published for this 2015 E.C. social item (Average rate of change). Distance-time table is only in the image; cannot compute average rates of change without the values.Try the options yourself, then practise the same subject timed in the app.

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2015 E.C. · social · Q52 · Higher-order derivatives

If ff and gg are functions defined by f(x)=x7−10x5+x3f(x)=x^7-10x^5+x^3 and g(x)=e−2xg(x)=e^{-2x} respectively, then which one of the following is true?
  1. A. f(n)(x)=0,∀n≥6f^{(n)}(x)=0, \forall n\geq 6
  2. B. g(n)(x)=(−1)n2ne−2x,∀n∈Ng^{(n)}(x)=(-1)^n 2^n e^{-2x}, \forall n\in\mathbb{N}
  3. C. (12f)(n)(x)=0,∀n≥6\left(\frac{1}{2}f\right)^{(n)}(x)=0, \forall n\geq 6
  4. D. (−g)(n)(x)=(−1)n2n+1e−2x,∀n∈N(-g)^{(n)}(x)=(-1)^{n}2^{n+1}e^{-2x}, \forall n\in\mathbb{N}
Worked solutionUnofficial

Answer: B

Answer B: g(n)(x)=(−1)n2ne−2x,∀n∈Ng^{(n)}(x)=(-1)^n 2^n e^{-2x}, \forall n\in\mathbb{N}. The stem is: If ff and gg are functions defined by f(x)=x7−10x5+x3f(x)=x^7-10x^5+x^3 and g(x)=e−2xg(x)=e^{-2x} respectively, then which one of the following is true? Letter B is the Abrari key for this Higher-order derivatives. g^(n)(x) = (-2)^n e^{-2x} = (-1)^n 2^n e^{-2x}. A/C are false since f is degree 7 (nonzero 7th derivative). B is true.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q55 · Derivatives - computation

Which one of the following is true about the derivative of the given functions?
  1. A. ddx(xsin⁡x)=cos⁡x−xsin⁡x\frac{d}{dx}(x\sin x)=\cos x-x\sin x
  2. B. ddx(x−ln⁡xx4)=−4x2+x−1+4xln⁡xx6,x\frac{d}{dx}\left(\frac{x-\ln x}{x^4}\right)=\frac{-4x^2+x-1+4x\ln x}{x^6}, x≠00
  3. C. If f(x)=cos⁡x1+sin⁡xf(x)=\frac{\cos x}{1+\sin x}, then ddx[f(x)]∣x=π=0\frac{d}{dx}[f(x)]|_{x=\pi}=0
  4. D. If h(x)=3xexh(x)=^3\sqrt{x}e^x, then ddx[h(x)]∣x=27=8227e27\frac{d}{dx}[h(x)]|_{x=27}=\frac{82}{27}e^{27}
Worked solutionUnofficial

Answer: D

Answer D: If h(x)=3xexh(x)=^3\sqrt{x}e^x, then ddx[h(x)]∣x=27=8227e27\frac{d}{dx}[h(x)]|_{x=27}=\frac{82}{27}e^{27}. The stem is: Which one of the following is true about the derivative of the given functions? Letter D is the Abrari key for this Derivatives - computation. h(x)=x^{1/3}e^x, h'(27) = (1/3)(27)^{-2/3}e^{27} + (27)^{1/3}e^{27} = (1/27)e^{27} + 3e^{27} = (82/27)e^{27}. D is true.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q58 · Integration - concept

Which one of the following is true about integration and differentiation?
  1. A. Integral of a function is an anti-derivative of its derivative.
  2. B. The process of finding f(x)f(x) from ∫f(x)dx\int f(x)dx is called integration.
  3. C. Integration is the reverse process of differentiation.
  4. D. Any continuous function has only one anti-derivative.
Worked solutionUnofficial

Answer: C

Answer C: Integration is the reverse process of differentiation.. The stem is: Which one of the following is true about integration and differentiation? Letter C is the Abrari key for this Integration - concept. C is the standard true statement. D is false (infinitely many anti-derivatives differing by a constant); B is worded backwards; A is imprecise.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q60 · Integration - trigonometric

Which one of the following integrals of trigonometric functions is the true?
  1. A. ∫(sin⁡2x)dx=12cos⁡2x+C\int(\sin2x)dx = \frac{1}{2}\cos2x+C
  2. B. ∫10(cos⁡10x)dx=−sin⁡10x+C\int10(\cos10x)dx = -\sin10x+C
  3. C. ∫5(cos⁡7x+3)dx=−57sin⁡7x+C\int5(\cos7x+3)dx = -\frac{5}{7}\sin7x+C
  4. D. ∫cos⁡(−5x)dx=15sin⁡(5x)+C\int\cos(-5x)dx = \frac{1}{5}\sin(5x)+C
Worked solutionUnofficial

Answer: D

Answer D: ∫cos⁡(−5x)dx=15sin⁡(5x)+C\int\cos(-5x)dx = \frac{1}{5}\sin(5x)+C. The stem is: Which one of the following integrals of trigonometric functions is the true? Letter D is the Abrari key for this Integration - trigonometric. cos(-5x)=cos(5x); ∫cos(5x)dx = (1/5)sin(5x)+C. D is true. A has wrong sign, B is +sin10x, C omits the +15x term.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q62 · Definite integrals - area

What is the area (in square units) of the regions bounded by the graph of the equation f(x)=2x+1,y=0,x=−1andx=4f(x) = 2x + 1, y = 0, x = -1 and x = 4?
  1. A. 41
  2. B. 20.5
  3. C. 40
  4. D. 21.5
Worked solutionUnofficial

Answer: B

Answer B: 20.5. The stem is: What is the area (in square units) of the regions bounded by the graph of the equation f(x)=2x+1,y=0,x=−1andx=4f(x) = 2x + 1, y = 0, x = -1 and x = 4? Letter B is the Abrari key for this Definite integrals - area. Area = ∫_{-1}^{4}|2x+1|dx = 0.25 (for x in [-1,-0.5]) + 20.25 (for [-0.5,4]) = 20.5.

Unofficial — solved by Abrari, not an official EAES key.

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2015 E.C. · social · Q63 · Definite integrals - area

What is the area (in square units) of the region enclosed by the curve f(x)=x2−4x+3f(x) = x^2 - 4x + 3, the x-axis, the vertical lines x=−1x = -1 and x=3x = 3?
  1. A. 8
  2. B. 12
  3. C. 5
  4. D. 17
Worked solutionUnofficial

Answer: A

Answer A: 8. The stem is: What is the area (in square units) of the region enclosed by the curve f(x)=x2−4x+3f(x) = x^2 - 4x + 3, the x-axis, the vertical lines x=−1x = -1 and x=3x = 3? Letter A is the Abrari key for this Definite integrals - area. Area = ∫_{-1}^{3}|x^2-4x+3|dx = 20/3 (on [-1,1]) + 4/3 (on [1,3]) = 8.

Unofficial — solved by Abrari, not an official EAES key.

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2016 E.C.

2016 E.C. · natural · Q29 · calculus

Which one of the following is true about the integral of a function f(x) with respect to a variable x?
  1. A. ∫f(x) dx\int f(x)\,dx is the set of all derivatives of f.
  2. B. ∫f′(x) dx\int f'(x)\,dx is the set of all derivatives of f.
  3. C. ∫f(x) dx\int f(x)\,dx is the set of all anti-derivatives of f.
  4. D. ∫f′(x) dx\int f'(x)\,dx is the set of all anti-derivatives of f.
Worked solutionUnofficial

Answer: C

Answer C: ∫f(x) dx\int f(x)\,dx is the set of all anti-derivatives of f.. The stem is: Which one of the following is true about the integral of a function f(x) with respect to a variable x? Letter C is the Abrari key for this calculus.

Unofficial — solved by Abrari, not an official EAES key.

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2016 E.C. · natural · Q54 · calculus

What is the area of the region bounded by the graph of f(x)=x3f(x) = x^3 and the x-axis on the interval [−4,2][-4, 2]?
  1. A. 68 square units
  2. B. 136 square units
  3. C. 60 square units
  4. D. 120 square units
Worked solutionUnofficial

Answer: A

Answer A: 68 square units. The stem is: What is the area of the region bounded by the graph of f(x)=x3f(x) = x^3 and the x-axis on the interval [−4,2][-4, 2]? Letter A is the Abrari key for this calculus.

Unofficial — solved by Abrari, not an official EAES key.

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2016 E.C. · natural · Q59 · calculus

Let ff be differentiable at aa. In the expression f(a+h)−f(a)h\frac{f(a+h) - f(a)}{h} for h≠0h \neq 0, as hh gets closer to zero from both directions, the expression approaches f′(a)f'(a), which is equal to:
  1. A. The gradient of the tangent line to the graph of ff at (a,f(a))(a, f(a)).
  2. B. The gradient of the vertical line to the graph of ff at (a,f(a))(a, f(a)).
  3. C. The gradient of the secant line to the graph of ff at (a,f(a))(a, f(a)).
  4. D. The gradient of the normal line to the graph of ff at (a,f(a))(a, f(a)).
Worked solutionUnofficial

Answer: A

Answer A: The gradient of the tangent line to the graph of ff at (a,f(a))(a, f(a)).. The stem is: Let ff be differentiable at aa. In the expression f(a+h)−f(a)h\frac{f(a+h) - f(a)}{h} for h≠0h \neq 0, as hh gets closer to zero from both directions, the expression approaches f′(a)f'(a), which is equal to: Letter A is the Abrari key for this calculus.

Unofficial — solved by Abrari, not an official EAES key.

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2016 E.C. · social · Q41 · Average rate of change

An enterprise found its profit by f(t)=t2−t+100f(t) = t^2 - t + 100, t in days. Which of the following is the average rate of change of its profit on 0≤t≤20 \leq t \leq 2?
  1. A. f(0)−f(2)2\frac{f(0) - f(2)}{2}
  2. B. f(2)−f(1)2\frac{f(2) - f(1)}{2}
  3. C. f(2)−f(t)t−2\frac{f(2) - f(t)}{t-2}
  4. D. 12(f(2)−f(0))\frac{1}{2}(f(2) - f(0))
Worked solutionUnofficial

Answer: D

Answer D: 12(f(2)−f(0))\frac{1}{2}(f(2) - f(0)). The stem is: An enterprise found its profit by f(t)=t2−t+100f(t) = t^2 - t + 100, t in days. Which of the following is the average rate of change of its profit on 0≤t≤20 \leq t \leq 2? Letter D is the Abrari key for this Average rate of change. Average rate = (f(2)-f(0))/(2-0).

Unofficial — solved by Abrari, not an official EAES key.

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2016 E.C. · social · Q43 · Integration

Which of the following gives the definition of integration for a function?
  1. A. It is a process of reversing the derivative of the function.
  2. B. It is the derivative of a function at a point.
  3. C. It is the rate of change of a function.
  4. D. It is the derivative of a function on the given interval
Worked solutionUnofficial

Answer: A

Answer A: It is a process of reversing the derivative of the function.. The stem is: Which of the following gives the definition of integration for a function? Letter A is the Abrari key for this Integration.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C.

2017 E.C. · natural · Q2 · derivatives

Let f(x)=(1−2x)12f(x) = (1 - 2x)^\frac{1}{2}. Then what is the values of f′(x)f'(x)?
  1. A. 11−2x\frac{1}{\sqrt{1 - 2x}}
  2. B. −121−2x\frac{-1}{2} \sqrt{1 - 2x}
  3. C. 121−2x\frac{1}{2} \sqrt{1 - 2x}
  4. D. −11−2x-\frac{1}{\sqrt{1 - 2x}}
Worked solutionUnofficial

Answer: D

Answer D: −11−2x-\frac{1}{\sqrt{1 - 2x}}. The stem is: Let f(x)=(1−2x)12f(x) = (1 - 2x)^\frac{1}{2}. Then what is the values of f′(x)f'(x)? Letter D is the Abrari key for this derivatives.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C. · natural · Q11 · extrema of functions

Consider the function f(x)=x3−3x2+3f(x) = x^3 - 3x^2 + 3 on (−2,3)(-2, 3). Then which of the following point is local minimum and local maximum of f(x)f(x) respectively?
  1. A. (−2,−27)(-2, -27) and (0,3)(0, 3)
  2. B. (0,3)(0, 3) and (2,−1)(2, -1)
  3. C. (3,3)(3, 3) and (−2,−27)(-2, -27)
  4. D. (2,−1)(2, -1) and (0,3)(0, 3)
Worked solutionUnofficial

Answer: D

Answer D: (2,−1)(2, -1) and (0,3)(0, 3). The stem is: Consider the function f(x)=x3−3x2+3f(x) = x^3 - 3x^2 + 3 on (−2,3)(-2, 3). Then which of the following point is local minimum and local maximum of f(x)f(x) respectively? Letter D is the Abrari key for this extrema of functions.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C. · natural · Q21 · average rate of change

What is the average rate of change of g(x)=x3−2xg(x) = x^3 - 2x over the interval 1≤x≤61 \leq x \leq 6?
  1. A. 2035\frac{203}{5}
  2. B. 41
  3. C. 375\frac{37}{5}
  4. D. 46
Worked solutionUnofficial

Answer: B

Answer B: 41. The stem is: What is the average rate of change of g(x)=x3−2xg(x) = x^3 - 2x over the interval 1≤x≤61 \leq x \leq 6? Letter B is the Abrari key for this average rate of change.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C. · natural · Q30 · derivatives

Suppose f(x)=3xf(x) = 3\sqrt{x} and g(x)=13x4+2xg(x) = \frac{1}{3}x^4 + 2x be given two functions. Then which of the following is equal to (f+g)′(x)(f + g)'(x)?
  1. A. 3x−12+4x3+23x^{-\frac{1}{2}} + 4x^3 + 2
  2. B. 23x−13+4x3+2x\frac{2}{3}x^{-\frac{1}{3}} + 4x^3 + 2x
  3. C. 23x−13+4x3+2\frac{2}{3}x^{-\frac{1}{3}} + 4x^3 + 2
  4. D. 32x−12+43x3+2\frac{3}{2}x^{-\frac{1}{2}} + \frac{4}{3}x^3 + 2
Worked solutionUnofficial

Answer: D

Answer D: 32x−12+43x3+2\frac{3}{2}x^{-\frac{1}{2}} + \frac{4}{3}x^3 + 2. The stem is: Suppose f(x)=3xf(x) = 3\sqrt{x} and g(x)=13x4+2xg(x) = \frac{1}{3}x^4 + 2x be given two functions. Then which of the following is equal to (f+g)′(x)(f + g)'(x)? Letter D is the Abrari key for this derivatives.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C. · social · Q17 · Calculus - derivative at stationary point

Let ff be a function and let cc be an element of domain of ff at which the gradient of ff is zero. Which one of the following statements is true about f′(c)f'(c)?
  1. A. f′(c)=0f'(c)=0
  2. B. f′(c)≠0f'(c)\neq 0
  3. C. f′(c)>0f'(c)>0
  4. D. f′(c)<0f'(c)<0
Worked solutionUnofficial

Answer: A

Answer A: f′(c)=0f'(c)=0. The stem is: Let ff be a function and let cc be an element of domain of ff at which the gradient of ff is zero. Which one of the following statements is true about f′(c)f'(c)? Letter A is the Abrari key for this Calculus - derivative at stationary point.

Unofficial — solved by Abrari, not an official EAES key.

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2017 E.C. · social · Q22 · Calculus - gradient

The gradient of a function f(x)=1x2f(x) = \frac{1}{x^2} at x=ax = a, a≠0a \neq 0 is equal to:
  1. A. −2a4-\frac{2}{a^4}
  2. B. −2a3-\frac{2}{a^3}
  3. C. 2a2\frac{2}{a^2}
  4. D. 2a3\frac{2}{a^3}
Worked solutionUnofficial

Answer: B

Answer B: −2a3-\frac{2}{a^3}. The stem is: The gradient of a function f(x)=1x2f(x) = \frac{1}{x^2} at x=ax = a, a≠0a \neq 0 is equal to: Letter B is the Abrari key for this Calculus - gradient.

Unofficial — solved by Abrari, not an official EAES key.

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