Mathematics Mock Board

Question: What is the third derivative of the function f(x) = 4x^3-6x^2+2x+8 A. 24x-12 B. 12x^2-12x+2 C. 24x^2-12x+2 D. 12x^2-12

A

Question: f(x, y) = sin(y + yx^2) / 1 + x^2 Value of fxy at (0,1) is A. 0 B. 1 C. 67 D. 90

A

Question: which of the following is not an application of derivatives A. To check the date. B. Derivatives are used to derive many equations in Physics. C. To check the temperature variation. D. To calculate the profit and loss in business using graphs.

A

Question: The parametric equations for a curve are given by x = θ − sin θ, y = 1 − cos θ. Find dx/dy as a function of θ. a. dx/dy = (1 − cos θ) / sin θ b. dx/dy = (1 + cos θ) / sin θ c. dx/dy = (cos θ - 1) / sin θ d. dx/dy = (cos θ + 1) / sin θ

A #6 https://www.sfu.ca/~vjungic/Zbornik2020/sec_Parametric_Curves.html

Question: If y = 4cosx + sin2x , what is the slope of the curve when x = 2 radians. A. -2.21 B. -4.94 C. -3.25 D. 2.21

B. -4.94

Question: Find an equivalent equation in rectangular coordinates r = cos θ: A. x2 + y2 = y B. x2 + y2 = x C. (x + y)2 = y D. (x + y)2 = x

B. x2 + y2 = x

Question: Consider the function f(x) = 3x^4 - 8x^3 + 6x^2 - 12x +5 f'''(x)= A. 72x + 48 B. 36x - 48 C. 36x + 48 D. 72x - 48

D. 72x - 48

Question: A voltage signal, V(t), is given by V(t) = 4t^3 - 12t^2 + 10t + 6. What is the expression for the current, I(t), when the voltage signal is applied to a resistor with resistance R? A. I(t) = (12t^2 - 24t + 10)/R B. I(t) = 12t^2 - 24t + 10 C. I(t) = 4t^3 - 12t^2 + 10t D. I(t) = (4t^3 - 12t^2 + 10t + 6)/R

A.

Question: Given the function f(x) = 3x^4 - 8x^3 + 6x^2 - 2x + 1, what is the third-order derivative of f(x)? A. f'''(x) = 72x - 48 B. f'''(x) = 12x^2 - 24x + 6 C. f'''(x) = 36x^2 - 48x + 6 D. f'''(x) = 6x^3 - 12x^2 + 6x - 2

A.

"Question: https://docs.google.com/document/d/17pcUd3Cs8Jz0KZ-uC6yi2ljK0v7J646tORLZ1ufVjfg/edit A. 1 B. 1/2 C. 1/sqrt e D. 0

A. 1

Question: If h'(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 2, what is the second derivative, h''(x), with respect to x? A. 12x^3 - 6x^2 + 10x - 7 B. 12x^3 - 6x^2 + 10x C. 6x^3 - 3x^2 + 5x - 7 D. 6x^3 - 3x^2 + 5x

A. 12x^3 - 6x^2 + 10x - 7

Question: For the function f(x) = 2x^2 - 3x + 1, what is the limit as x approaches 2? A. 3 B. 4 C. 5 D. 6

A. 3

Question: Which of the following functions is continuous at x = 3? A. f(x) = 2x^2 - 1 B. g(x) = (x^2 - 9)/(x - 3) C. h(x) = 1/(x - 3) D. j(x) = |x - 3|

A. f(x) = 2x^2 - 1

Question: If the derivative of a function f(x) is f'(x) = 2x + 5, what is the original function f(x)? A. f(x) = x^2 + 5x + C (where C is a constant) B. f(x) = 2x^2 + 5x + C C. f(x) = 2x + 5 D. f(x) = x^2 + 5

A. f(x) = x^2 + 5x + C (where C is a constant)

Question: If f(x)=x3−3x2+2x−1, what is f′(2)? A.2 B.4 C.6 D.8

A.2

Question: If f(x)=x4−2x3+x2−x+1, what is the second derivative of f at x=1, i.e., f′′(1)? A.2 B.4 C.6 D.8

A.2

Question: Calculate the second derivative of the function h(x) = 4x^4 - 2x^3 + 7x^2 - 1. A. 16x^3 - 6x^2 + 14x B. 48x^2 - 12x + 14 C. 8x^4 - 6x^3 + 14x^2 D. 4x^3 - 6x^2 + 7x

B

Question: Differentiate cos(sin x) with respect to x: A. -sin x ∙ cos (cos x) B. -cos x ∙ sin (sin x) C. cos x ∙ sin (sin x) D. sin x ∙ cos (sin x)

B

Question: Evaluate the limit of the given. https://docs.google.com/document/d/1aaLRL_VT6PPjGboy_MsTGb8AmBzrrOXN3XnicY6Z0Hs/edit?usp=sharing A. 1/5 B. 2/5 C. 3/5 D. 4/5

B

Question: Find a set of parametric equations for the equation y = x^2 + 5. A. x = t^2 + 5; y = t B. x = t; y = t^2 + 5 C. x = t; y = t + 5 D. x = t^2; y = t^+5

B

Question: Find the indicated derivative of the function d2y/dx2 for y = 7x sin x A) 7 cos x - 14x sin x B) 14 cos x - 7x sin x C) -14 cos x + 7x sin x D) -7x sin x

B

Question: Find the second derivative of f(x)=(5x^4)-(3x^3)+(7x^2)-9x+2. A. f''(x)=(20x^3)-(9x^2)+14x-9 B. f''(x)=(60x^2)-18x+14 C. f''(x)=(60x^2)+18x-14 D. f''(x)=(20x^3)+(9x^2)-14x+9

B

Question: Find the second derivative of y = x^(-2) at x = 2. A. 96 B. 0.375 C. -0.25 D. -0.875

B

Question: Find the value of dy/dx if x = cos t, y = sin t. A. -tan t B. -cot t C. cot t D. tan t

B

Question: If f is a continuous function on a real line given that f(x) = x^2+4x+9, then the value of f(7) is. A. 96 B. 86 C. undefined D. 91

B

Question: Consider the parametric equations for a curve: x(t) = 3t^2 + 2t y(t) = t^3 - t Find the equation of the tangent line to the curve at the point where t = 2. A. y = 12x - 22 B. y = (14/11)x - (158/11) C. y = 45x - 96 D. y = (5/2)x - (6/5)

B.

Question: Consider the parametric equations x=t^2 and y=t^3−t.What is the derivative dy/dx at t=1? A.0 B.1 C.2 D.3

B. 1

Question: The second-order derivative for y = 7x sin x is: A. 7 cos x - 14x sin x B. 14 cos x - 7x sin x C. -14 cos x + 7x sin x D. -7x sin x

B. 14 cos x - 7x sin x

Question: What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2? A. 2 B. 4 C. 0 D. Undefined

B. 4

Question: Find the second derivative of x3 - 5x2 + x = 0. A. 10x - 5 B. 6x -10 C. 3x + 10 D. 3x2 - 5x

B. 6x - 10

Question: Determine the derivate for the following with respect to x, f(x) = (4x^3 - 7x + 8)/x A. (12x^2 )- 7 B. 8x−8x^(-2) C. 8x-7 D. 12x^2 - 8x +7

B. 8x−8x^(-2)

Question: If f(x) = g(u) and u = u(x) then A. f '(x) = g '(u) B. f '(x) = g '(u) . u '(x) C. f '(x) = u '(x) D. None of the above

B. f '(x) = g '(u) . u '(x)

Question: Find the partial derivatives with respect to x of the function xy2 -5y + 6. A. y2 - 5 B. y2 C. xy - 5y D. 2xy

B. y2

"Question: Evaluate as x approaches infinity. lim ((x^2)/(1+x^2) A. 0 B. 1/2 C. 1 D. 2

C

Question: Determine the fourth derivative of h(t)=3t^7-6t^4+8t^3-12t+18 A. 126t^5-72t^2+48t B. 630t^4-144t+48 C. 2520t^3-144 D. 7560t^2

C

Question: Find The slope of the tangent to the curve y = 9x^2+7x^4 at x=1 A. 28 B. 16 C. 46 D. None of the above

C

Question: Given the function f(x, y) = 2x^2y - 3y + 5x, what is the partial derivative of f with respect to x? A. 4xy - 3 B. 2xy - 3 C. 4xy + 5 D. 2xy + 5

C

Question: If f(x) = 4x^3 - 2x^2 + 7x - 1, what is the second derivative, f''(x), with respect to x? A. 24x - 4 B. 12x^2 - 4x + 7 C. 24x^2 - 4 D. 12x^3 - 4x^2 + 7x

C

Question: In a parametric equation system, if x(t) = 2t^2 + 3t and y(t) = 3t - 1, what is the equation of the curve represented by these parametric equations? A. y = 2x - 5 B. y = x^2 + 2x - 1 C. y = x^2 + 2x D. y = 3x - 1

C

Question: Use Simpson's Rule with n=6 to estimate the length of the curve x=t-e^t, y=t+e^t, -6≤t≤6. A. 720.3053 B. 702.3053 C. 612.3053 D. 600.3053

C

Question: Which of the following functions is continuous for all real numbers? A. h(x)= 1/x B. h(x)= IxI C. h(x)= sqrt 3 D. h(x)= 1/x-3

C

Question: https://drive.google.com/drive/folders/1y0oD6htiflL8diF_0pEAB7C2iPmEW207?usp=sharing A. 2 B. 4 C. 8 D. ½

C

Question: Given the parametric equations: x = 2t^2 + 3t y = t - 1 Which of the following options represents the Cartesian equation of the respective curve? A. y = 2x^2 + 3x - 1 B. y = 4x^2 + 6x - 1 C. y = 4x^2 - 6x - 1 D. y = 2x^2 - 3x - 1

C (y = 4x^2 - 6x - 1)

Question: Consider the function f(x) = (x^2 - 4) / (x - 2). Determine whether the function is continuous at x = 2. A. The function is continuous at x = 2. B. The function is not continuous at x = 2, but has a removable discontinuity. C. The function is not continuous at x = 2, and it has an essential discontinuity. D. The function is not continuous at x = 2, but it has a jump discontinuity.

C.

Question: Given the function f(x) = 4x^5 - 10x^3 + 2x - 8, what is the fourth derivative of the function f(x)? A. f''''(x) = 240x^2-60 B. f''''(x) = 8x^4-30x^2+2 C. f''''(x) = 480x D. f''''(x) = 20x^4-30x^2+2

C.

Question: A curve C is given by the parametric equation x = 2t^2 - 1, y=3(t+1), t ∈ R Determine the coordinate of the points of intersection between C and the straight line with equation. 3x - 4y =3 A. (12,7) & (1,0) B. (7,12) & (0,1) C. (17,12) & (1,0) D. (19, 14) & (8,9)

C. (17,12) & (1,0)

Question: Let f(x)=x^2-4/x-2, for all x is not equal to 2. What is the limit of f(x) as x approaches 2? A.0 B.2 C.4 D.Undefined

C. 4

Question: Find the Derivative of the polynomial below: (ax + 1)2 - 2ax3; a = const A. 4ax^2 - 2a^3(x) B. 2ax^2 + 2ax C. 6ax^2 + 2a^2(x) + 2a. D. 3ax^2 + a^2(x) + a.

C. 6ax^2 + 2a^2(x) + 2a.

Question: Evaluate the lim (x2 -16 )/(x-4) such that x approaches to 4. A. 0 B. 1 C. 8 D. 16

C. 8

Question: Suppose that f,g are continuous functions on R. Which of the following is FALSE? A. f(g(x)) is well defined for all real numbers, and is continuous on R. B. limx→af(x)g(x) exists for any real number a. C. f(x)g(x) has either an absolute maximum, absolute minimum, or both. D. f(x)/g(x) may not be well defined for all real numbers.

C. f(x)g(x) has either an absolute maximum, absolute minimum, or both.

Question: Define g(x) = x^3 - 9x^2 + 24x + 17. Give the interval(s) on which g is decreasing. A. (3, ∞) B. (-∞, 3) C. (-∞, 2) ∪ (4, ∞) D. (2, 4)

D

Question: Find the second derivative of the following functions with respect to x. y=3x2+5x-1 A. 6x+5 B. 5 C. 3x+5 D. 6

D

Question: If log10 (x - 9) + log10 x = 1 then the value of x is A. 0 B. 1 C. 5 D. None of the above

D

Question: What is the derivative with respect to x of (x + 1)^3-x^3? A. 3x + 6 B. 3x - 3 C. 6x - 3 D. 6x + 3

D

Question: What is the value of lim(y→2) (y^2 - 4y - 2)? a) 2 b) 4 c) 1 d) -6

D

Question: Which of the following is NOT a partial derivative rule? A. Product Rule B. Quotient Rule C. Power Rule D. Zero Identity Rule

D

Question: Which of the following statements is always true ([.] represents the greatest integer function)? A. If f(x) is discontinuous, then |f(x)| is discontinuous B. If f(x) is discontinuous, then f(|x|) is discontinuous C. f(x) = [g(x)] is discontinuous when g(x) is an integer D. None of the above

D

Question: f(x, y) = sin(xy) + x2 ln(y) Find fyx at (0, π⁄2) A. 33 B. 0 C. 3 D. 1

D

Question: if x = t^2 and y = t^3, then d²y/dx² = ? A. 0 B. t C. √t D. 3/4t

D

Question: For the function f(x) = (x^3 - 4x^2 + 5x - 2) / (x^2 - 3x + 2), what is the limit of f(x) as x approaches 1? A. 0 B. -∞ C. 1 D. Does not exist

D (Does not exist)

Question: If u and v are continuous on [a,b] and have equal finite derivatives in [a,b], then u - v is ________. A. infinite B. f/g C. g D. constant

D.

Question: What is the derivative with respect to x of (x+1)^3 - x3 ? A. 3x + 6 B. 3x - 3 C. 6x - 3 D. 6x + 3

D. 6x + 3

Question: Find the second order derivative of y = 9 log⁡(t^3). a) 27/t b) -27/t^2 c) -1/t^2 d) -27/2t^2

ahttps://www.sanfoundry.com/mathematics-questions-Answers-second-order-derivatives/ Explanation: Given that, y=9 log⁡t3 dydx=9.1t3.3t2=27t d2ydx2=27(−1t2)=−27t2.

Question: Find f_x at (1, 1, 1) for the function f(x, y) = x^2 + xyz + z. a) 0 b) 1 c) 3 d) -1

c https://www.sanfoundry.com/engineering-mathematics-questions-Answers-partial-differentiation-1/ Explanation: fx = 2x + yz Put (x,y,z) = (1,1,1) fx = 2 + 1 = 3.

Question: What is the slope of the tangent to the curve y = 2x/(x^2 + 1) at (0, 0)? a) 0 b) 1 c) 2 d) 3

c https://www.sanfoundry.com/mathematics-questions-Answers-application-derivative/ Explanation: We have y = 2x/(x2 + 1) Differentiating y with respect to x, we get dy/dx = d/dx(2x/(x2 + 1)) = 2 * [(x2 + 1)*1 - x * 2x]/(x2 + 1)2 = 2 * [1 - x2]/(x2 + 1)2 Thus, the slope of tangent to the curve at (0, 0) is [dy/dx](0, 0) = 2 * [1 - 0]/(0 + 1)2 Thus [dy/dx](0, 0) = 2.

"Question: If f(x) = e^sin (log cos x) and g(x) = log cos x, then what is the derivative of f(x) with respect to g(x)? A. f(x) cos [g(x)] B. f(x) sin [g(x)] C. g(x) cos [f(x)] D. g(x) sin [f(x)]

A

Question: At what point does the function g(x) = x^3-6x^2+9x+4 have a local minimum? A. x = 3 B. x = 2 C. x = 4 D. x = 1

A

Question: Consider the functions I. e^-x II. x^2 - sin x III. sqrt(x^3+1) Which of the above functions is/are increasing everywhere in [0, 1]? A. III B. II C. I and III D. II and III

A

Question: Evaluate the following limits https://drive.google.com/file/d/13_HfrjSykgae1CegfEtUFAoRU44YJUZU/view?usp=sharing A. 39 B. 38 C. 40 D. 42

A

Question: Find out the parametric equation of a parabola (x - 3) = -16(y - 4). A) x = 3 - 8t, y = 4 - 4t2 B) x = 8t, y = 4t2 C) x = 2, y= t^2 D) x = -4. y= 2t

A

Question: Find the derivative of the function g(x) = 3x^3 - 2x^2 + 5x - 1 with respect to x. A. 9x^2 - 4x + 5 B. 12x^2 - 4x + 5 C. 3x^4 - 2x^3 + 5x^2 D. 3x^2 - 2x + 5

A

Question: Find the length of one arc of the curve whose parametric equations are x = 2θ - 2sin θ and y = 2 - 2cos θ. A. 16 B. 18 C. 14 D. 12

A

Question: Find the partial derivative of the function f(x, y) = 2x^3y - 5xy^2 with respect to x. A. 6x^2y - 5y^2 B. 2x^3 - 5y^2 C. 3x^2y - 5y^2 D. 6x^2 - 5xy

A

Question: Find the partial derivative with respect to x of the function xy^2- 5y +6. A. y^2 B. 6x-10 C. 3x+10 D. 3x^2-5x

A

Question: Find the second derivative for the function 3x^5 + 2x^3 - 6x +4 A. 60x^3 +12x B. 15x^4 + 6x^3 -6 C. 3x + 10 D. 40x^2 +2x

A

Question: Find the standard form of the equation of the parabola using the information given. 1 ) Focus: (-7, -1); Directrix: x = 1 1 ) A) (y + 3)2 = -1 6(x + 1) B) (x + 3)2 = -1 6(y + 1) C) (x + 1)2 = -1 6(y + 3) D) (y + 1)2 = -1 6(x + 3 D) (y + 1)2 = -1 6(x + 3)

A

Question: For the function h(x, y, z) = x^2y - z^3, calculate ∂h/∂z: A) ∂h/∂z = -3z^2 B) ∂h/∂z = x^2y - 3z^2 C) ∂h/∂z = 2xy D) ∂h/∂z = -x^2y

A

Question: For the function h(x,y) = e^xy+x^3y, what is ∂h/∂x ? A. ye^xy + 3x^2y B. e^xy + 3xy C. ye^xy + 3x^2 D. xe^xy +3x^2y

A

Question: For the parametric equations x= sin (t) and y= cos(t), what is the relationship between x and y that describe the curve A. x^2+y^2 = 1 B. x+y = 1 C. x^2-y^2 = 1 D. x-y = 1

A

Question: Given that 𝑓(𝑥) = 2𝑥^4 − 3𝑥^3 + 4𝑥^2, what is 𝑓′′′(𝑥)? A. 𝑓'''(𝑥) = 48𝑥 - 18 B. 𝑓'''(𝑥) = -48𝑥 - 18 C. 𝑓'''(𝑥) = 48𝑥 + 18 D. None of these

A

Question: Given the parametric equations x = 2t and y = t^2, find the coordinates of the point where the curve intersects the y-axis. A. (0, 0) B. (1, 1) C. (2, 4) D. (3, 9)

A

Question: If f(x) = 3x^4 - 2x^3 + 5x^2, what is the derivative f'(x) with respect to x? A. 12x^3 - 6x^2 + 10x B. 9x^3 - 6x^2 + 5x C. 12x^3 - 6x^2 + 5x D. 9x^3 - 4x^3 + 10x

A

Question: If y = 5 cos x - 3 sin x , then its 2nd order derivative is equal to : A. -y B. y C. 25y D. 9y

A

Question: In multivariable calculus, the partial derivative of a function f(x, y) with respect to the variable x is denoted by: A) ∂f/∂x B) ∂f/∂y C) df/dx D) δf/δx

A

Question: The parametric equations for a curve are given by x = 2θ + cos θ, y = θ^2 - sin θ. Find dx/dy as a function of θ. a. dx/dy = (2 - sin θ) / (2θ - cos θ) b. dx/dy = (2 + sin θ) / (2θ - cos θ) c. dx/dy = (2 - cos θ) / (2θ + sin θ) d. dx/dy = (2 + cos θ) / (2θ - sin θ)

A

Question: The side of an equilateral triangle is increasing at the rate of 2 cm/s. The rate at which area increases when the side is 10 is A) 10√3 cm2/s B) 10/3 cm2/s C) √3 cm2/s D) 10 cm2/s

A

Question: Which of the following options correctly represents the derivative of a function f(x) with respect to x at a given point? A. f'(x) B. f(x) C. ∫ f(x) dx D. ∂f/∂x

A (f'(x))