differential calculus

• Derivative


  The rate of change of a function with respect to one of its variables
• Finding the derivative of y = X^x

  
  

  C. x^x(1 + ln x)
• Calculate the lim (1/x+1/2)/(x^3+8) as x approaches 2

  1. A. -1/48

  2. B. 3/28

  3. C. 5/26

  4. D. Can't solve
• Consider the function f(x) = {a+bx, if x>2; 3, if x=2; b-ax^2, if x<2. Determine the values of constants a and bsuch that lim f(x) as x approaches 2, exists and is equal to f(2).

  1. A. -1/3 and 5/3

  2. B. 2/3 and -1/3

  3. C. 2/3 and 4/3

  4. D. 4/3 and -1/3
• A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper. Find the minimum length of the resulting crease.

  1. A. 6.79 in.

  2. B. 7.79 in.

  3. C. 8.79 in.

  4. D. 6.87 in.
• Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What time t does the minimum distance between Car A and Car B occur?

  1. A. 0.23 min.

  2. B. 24.96 min.

  3. C. 13.8 min.

  4. D. 20.77 min.
• . Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence.

  1. A. 17.64 ft.

  2. B. 8.77 ft.

  3. C. 16.67 ft.

  4. D. 14.99 ft.
• Finding the slope of the curve y = 2 (1 + 3x)^2 at point (0,3)

  1. A. 9

  2. B. 12

  3. C. 0.16

  4. D. 0.21
• If the slope of the curve y^2 = 12x is equal to 1 at point (x,y) find the value of x.

  1. A. 6

  2. B. 3

  3. C. 4

  4. D. 12
• Find the second derivative of y = 2x + 3(4x + 2)^3 when x=1

  1. A. 650

  2. B. 560

  3. C. 1278

  4. D. 1728
• A closed cylindrical can must have a volume of 1000 in3 . Its lateral surface is to be constructed from a rectangular piece of metal and its top and bottom are to be stamped from square pieces of metal and the rest of the square discarded. What height will minimize the amount of metal needed in the construction of the can?

  1. A. 30/π in.

  2. B. 40/π in.

  3. C. 50/π in.

  4. D. 60/π in.
• . A closed cylindrical can must have a volume of 1000 in3 . What radius will minimize its surface area?

  1. A. 10.84 in.

  2. B. 7.87 in.

  3. C. 5.42 in.

  4. D. 12.67 in.
• . A publisher wants to print a book whose pages are each to have an area of 96 in2 . The margins are to be 1 in on each of three sides and 2 in on the fourth side to allow room for binding. What dimensions will allow the maximum area of for the printed region?

  1. A. 24 in x 4 in

  2. B. 16 in x 6 in

  3. C. 12 in x 8 in

  4. D. 19.2 in x 5 in
• . A cylindrical boiler is to have a volume of 1340 cu. ft. The cost of the metal sheets to make the boiler should be minimum. What should be its base diameter in feet?

  1. A. 11.95 ft.

  2. B. 13.50 ft.

  3. C. 14.25 ft.

  4. D. 10.55 ft.
• A closed cylindrical tank has a capacity of 576.56 m3 . Find the minimum surface area of the tank.

  1. A. 383.4 m^2

  2. B. 338.4 m^2

  3. C. 218.4 m^2

  4. D. 128.4 m^2
• . If the sum of the two numbers is 4, find the minimum value of the sum of their cubes.

  
  

  A. 8

  B. 9

  C. 16

  D. 27
• Let f(x, y) = (x^2)(y^3). Determine (𝜕^2f)/(𝜕y𝜕x)

  
  

  A.2y^3

  B. 6xy^2

  C. 6yx^2

  D. 2x^3
• Let f(x, y) = (x^2)(y^3). Determine (𝜕^2f)/(𝜕y^2)

  1. A.2y^3
  2. B. 6yx^2
  3. C. 6xy^2
  4. D. 2x^3
• . Find the point on the line 3x+y=6 closest to (2,3).

  1. A.(10/11, 27/10)
  2. B. (11/10, 27/10)
  3. C. (7/10, 11/10)
  4. D. (10/11, 10/27)
• . Water is being pumped into a conical tank at the rate of 100 ft^3/min. The height of the tank is 20 ft and its radius is 5 ft. How fast is the water level rising when the water height is 10 ft?

  A. 5.09 ft/min

  B. 7.09 ft/min

  C. 6.09 ft/min

  D. 8.09 ft/min
• Evaluate ∫ 0 to 2 ∫ 0 to 3 ∫ 0 to 2 𝑑𝑧𝑑𝑦𝑑𝑥

  A. 24

  B. 12

  C. 15

  D. 20
• Evaluate 4 ∫ 0 to 2 ∫ 0 to sqrt(4-x^2) ∫ 0 to (4-x^2-y^2) 𝑑𝑧𝑑𝑦𝑑𝑥

  A. 8π

  B. 10π

  C. 5π

  D. 12π
• An object is projected vertically upward. If the distance travelled by an object is expressed as ℎ = 100𝑡 − 16.1𝑡 2 , what is the velocity of the object after 2 seconds?

  A. 33.2 m/s

  B. 35.6 m/s

  C. 32.2 m/s

  D. 22.8 m/s
• Find the radius of curvature at any point of the curve 𝑦 + ln cos 𝑥 = 0

  A. cos x

  B. 1.211

  C. sec x

  D. 1
• lim𝑥→1 (2 − 𝑥)^ (𝑡𝑎𝑛(��𝑥)/(2))

  A. 𝑒^2𝜋

  B. 0

  C. 𝑒^(2/𝜋)

  D. does not exist
• Lim𝑥→0 sin ( 1 /𝑥 )

  A. 0

  B. ½

  C. -1

  D. does not exist
• lim𝑥→0 (tan 2𝑥 −2 sin 𝑥)/(𝑥^3)

  A. 0

  B. infinity

  C. 3

  D. does not exist
• Lim 𝑥→𝜋/2 (𝑡𝑎𝑛𝑥 tan 2𝑥)

  A. 1

  B. -2

  C. 2

  D. 3
• Find the derivative of 𝑥 = 3𝑦^4 + 7𝑦^3 + 1 with respect to x.

  A. 1 /(12𝑦^3+21𝑦^2)

  B. 1 /(6𝑦^3+21𝑦^2)

  C. 1 /(12𝑦^3+18𝑦^2)

  D. 1 /(13𝑦^3+21𝑦^2)
• What is the derivative of 2 cos(2 + 𝑥^3 ) with respect to x?

  A. −6𝑥^2 sin (2 + ��^3 )

  B. 6𝑥^2 sin (2 + 𝑥^3 )

  C. 3𝑥^2 sin (2 + 𝑥^3 )

  D. −3𝑥^2 sin (2 + 𝑥^3 )
• Determine the slope of the curve 𝑥^2 + 𝑦^2 − 6𝑥 + 10𝑦 + 5 = 0 at the point (1, 0) A. 1/5

  B. 3/5

  C. 2/5

  D. 4/5
• Evaluate lim𝑥→3 (|x-5| +x)^5

  A. 3000

  B.3125

  C. 1267

  D. 4523
• Evaluate lim𝑥→2 (x^3 - 9)/(𝑥^2 - 8x +15)

  A. -5/3

  B. 5/3

  B. -8/3

  D. 9/3
• Evaluate lim𝑥→1 (10x - 9 -x^2)/(𝑥^2 - 1)

  A. 10

  B. 9/80

  C. 8

  D. 8/34
• Evaluate lim𝑥→1 (2-SQRT(4+2x))/(𝑥)

  A. 0.5

  B. 1

  C. – 0.5

  D. -1
• Evaluate lim𝑥→0^- (5x + sinx)/(𝑥)

  A. 6

  D. 5.5

  c. 3

  D. 8
• Evaluate lim𝑥→5^+ (X+8)/(𝑥-3)

  A. 6.3

  B. 5.1

  C. 7.5

  D. 6.5
• Evaluate lim𝑥→3^- (sqrt(x))/(𝑥-5)^2

  
  

  A. sqrt(3)/4

  B. - sqrt(3)/4

  C. sqrt(5)/3

  D. - sqrt(5)/3
• Evaluate lim𝑥→3 (4f(x) - 7h(x); f(x) = -6, h(x) = 2

  A. 41

  B. -23

  C. -38

  D. 17
• Evaluate lim𝑥→-3 (17-9x+x^4)

  A. 172

  B. 117

  C. 96

  D. 125