Module 08: Differential Calculus – Mechanical Engineering Online Review Center

Handouts
Exercises
Functions
Limit Concept
Maxima and Minima

LINK: Differential Calculus

Topics:

Differentiation of algebraic functions using formulas
Implicit differentiation
Higher derivatives Derivatives of Transcendental Functions
Maxima and minima

Find the derivative of the (3 – x²)^0.5
- a. x/(3-x²)^0.5
- b. 2x/(3-x²)^0.5
- c. x/(3x²-1)^0.5
- d. -x/(1-2x²)^0.5
Find the derivative of the 3(x² -1)^-1
- a. -6x/(x²-1)²
- b. 6x/(x²-2)²
- c. 8x/(x²-1)²
- d. -8x/(1-x²)²
Find the derivative of (4-x)/(x²-2)
- a. (x²-8x+2)/(x²-2)²
- b. (x²+8x-2)/(x²-2)²
- c. (x²+8x+2)/(x²-2)²
- d. (x²-8x-2)/(x²-2)²
Find the derivative of 4x² + 8x + 10.
- a. 7x+8
- b. 7x-8
- c. 8(x+3)
- d. 8(x+1)
Evaluate: lim_Δx→0 (x³-21)/(x-4)
- a. 32
- b. 40
- c. 48
- d. 52
Find the radius of curvature of the equation y = 4x³ – 4x² + 4x – 2 at point (3, 6).
- a. 11,312
- b. 12,091
- c. 10,650
- d. 9,012
Evaluate the limits (4x-1)/(10x+7) as x approaches infinity.
- a. 3/5
- b. 2/5
- c. 4/5
- d. 5/3
Find the derivative of y = (1-3x²)/(x-3) at x=1.
- a. 2.5
- b. 4.0
- c. 3.0
- d. 3.5
Find the partial derivatives with respect to x: x² + 6x + 10z².
- a. 2x+6
- b. 3x+2
- c. 2x-8
- d. 4x+3
Find the partial derivatives with respect to y: 5y² – 6y + 8
- a. 8y+6
- b. 10y-6
- c. 12y+3
- d. 6y-9
Find the partial derivatives with respect to x: 3x² – 4xy
- a. 6x+5y
- b. 8x+3y
- c. 6x-4y
- d. 6x-5y
Find the equation of the line tangent to the curve y = x + 3x^1/4 at point (9, 13).
- a. 8x+6y+20=0
- b. 8x-7y+19=0
- c. 9x+7y+21=0
- d. 9x-7y+18=0
What is the derivative with respect to x of (x + 2)³ – x³?
- a. 12(x+1)
- b. 14(x-1)
- c. 12(x2+2)
- d. 14(x-3)
Differentiate y = sec(x² + 3)
- a. 3xsec(x²+3)tan(x²+3)
- b. 2xsin(x²+3)cos(x²+3)
- c. -3xsec(x²+3)tan(x²+3)
- d. 2xsec(x²+3)tan(x²+3)
Find the derivative of f(x) = [x⁴ – (x – 1)³]³?
- a. 3[x⁴-(x-2)³]²(4x³-3x²+6x-4)
- b. 3x⁴-(x+2)³]²(4x³+3x²+6x-4)
- c. 3[x⁴+(x-3)³]²(4x³-5x²+6x-4)
- d. 3[x⁴-(x-1)³]²(4x³-3x²+6x-4)
What is the slope of the graph y = -3x² at the point (1, 4)?
- a. -8
- b. -5
- c. -6
- d. -4
The motion of a particle along a straight line is described by the equation x(t) = t³ – 4t² – 40t + 60 where x is expressed in feet and t in seconds. Compute the acceleration of the particle at the time in which v(t) = 0.
- a. 23.32 fps
- b. 28.27 fps
- c. 32.09 fps
- d. 20.12 fps
The motion of a particle is described by the equation x(t) = 3t³ – 10t² – 45t + 50, where x is measured in meters and t is measured in seconds. Find the velocity when the acceleration of the particle is equal to zero.
- a. -48.11 m/s
- b. -52.21
- c. -61.28
- d. -56.11
What is the slope of the line tangent to the parabola y = 14x² + 4 at point where x = 2?
- a. 56
- b. 52
- c. 58
- d. 50
The position of the acceleration of the object as a function of time is described by: x = 4t³ + 3t² – t + 4. What acceleration of the object at t = 3?
- a. 75
- b. 78
- c. 82
- d. 68
At what value of y does the inflection point occur for the curve y = 4x³ – 5x² – 25x + 26?
- a. 12
- b. 18
- c. 20
- d. 15
At which value of y does the relative minimum occur for the curve y = 3x³ – x² – 20x + 20.
- a. -3.22
- b. -4.39
- c. -2.27
- d. -1.88
The sum of two positive numbers is 60. What are the numbers if their product is to be the largest possible.
- a. 40 and 20
- b. 35 and 25
- c. 45 and 15
- d. 30 and 30
If the radius of a circle increases at the rate of 0.2 in/sec, find the rate of change of area when the radius is 5 inches long.
- a. 3π
- b. 2π
- c. 6π
- d. 4π
Find the maximum area of a rectangle whose perimeter is 150 in.
- a. 1,310.23 in²
- b. 1,406.25 in²
- c. 1,620.21 in²
- d. 1,812.14 in²
Find the maximum area of triangle whose perimeter is 40 in.
- a. 76.98 in²
- b. 83.14 in²
- c. 85.14 in²
- d. 74.11 in²
A spherical balloon is being filled with a rate of 2.4 cubic foot per second. Compute the time rate of change of the surface area of the balloon at the instant when the volume is 130 ft³.
- a. 2.3 ft²/sec
- b. 3.2 ft²/sec
- c. 1.5 ft²/sec
- d. 4.8 ft²/sec
A particle moves according to the following functions of time: x(t) = 3sin t, y(t) = 2cos t. What is the resultant velocity at t =π?
- a. 5.0
- b. 4.0
- c. 3.0
- d. 2.0
Given the function f(x) = 2x³ – 4x – 6, the minimum value of the function is:
- a. (3)^0.5/3
- b. -(5)^0.5/3
- c. (8)^0.5/3
- d. (6)^0.5/3
The value of x which provides the minimum y in the function: y = 3x³ – 20x + 16 is:
- a. 1.49
- b. 2.12
- c. 3.33
- d. 4.18
What is the radius, r, and height, h, of cylindrical oil can that holds 8 liters of oil but must have minimum surface area?
- a. 11.8 cm, 28.1 cm
- b. 10.8 cm, 21.7 cm
- c. 12.1 cm, 24.8 cm
- d. 9.2 cm, 25.1 cm
The product of two positive numbers is 18. Find the number if the sum of one and square of the other is least.
- a. 9 and 2
- b. 6 and 3
- c. 18 and 1
- d. 12 and 6
Differentiate y = e^x cos x³
- a. e^x(cosx³ – 3xsinx³)
- b. 2e^x(cosx³ + 3xsinx³)
- c. e^x(sinx³ – 3xcosx³)
- d. -e^x(2cosx³ – 3xsinx³)

Answer Key (Handwritten)