Calculus I: Master Differential and Integral Calculus for Mechanical Engineering

Question 1

Given a differentiable function f(x) on the interval [0, 1], if its derivative f'(x) is positive for all x in (0, 1), what can you conclude about the behavior of f(x) on the interval (0, 1)?

Accepted Answer

f(x) is increasing on the interval (0, 1).

Answer

f(x) has a minimum at x = 0.

Answer

f(x) is concave up on the interval (0, 1).

Answer

f(x) has an inflection point at x = 1.

Question 2

Determine the derivative of f(x) = x^3 - 2x^2 + 5x - 7 using the limit definition of the derivative.

Accepted Answer

f'(x) = 3x^2 - 4x + 5

Answer

f'(x) = 3x^2 + 4x - 5

Answer

f'(x) = 2x - 4

Answer

f'(x) = 5

Question 3

Calculate the derivative of the function f(x) = x³ - 2x² + 1.

Accepted Answer

3x² - 4x

Answer

x³ - 4x

Answer

3x² + 4x

Answer

x³ + 4x

Question 4

Which of the following is an application of the integral, but not the derivative?

Accepted Answer

Finding the volume of a solid of revolution

Answer

Finding the slope of a tangent line to a graph

Answer

Finding the velocity of an object moving along a curve

Answer

Finding the critical points of a function

Question 5

Determine the derivative of the function $f(x) = x^3 + 2x^2 - 5x + 1$.

Accepted Answer

$f'(x) = 3x^2 + 4x - 5$

Answer

$f'(x) = 3x^2 - 4x + 5$

Answer

$f'(x) = x^3 + 4x - 5$

Answer

$f'(x) = x^3 + 2x - 5$

Question 6

Determine the derivative of the function f(x) = (x³ - 2x² + x + 1)⁴.

Accepted Answer

4(x³ - 2x² + x + 1)³(3x² - 4x + 1)

Answer

(x³ - 2x² + x + 1)⁴

Answer

4x(x³ - 2x² + x + 1)³

Answer

(3x² - 4x + 1)

Question 7

Find the derivative of the function f(x) = (x² + 2x - 1) / (x - 1) using the quotient rule.

Accepted Answer

(2x + 2) / (x - 1) - (x² + 2x - 1) / (x - 1)²

Answer

(x² + 2x - 1) / (x - 1)²

Answer

(2x + 2) / (x - 1)

Answer

1 / (x - 1)

Question 8

Calculate the derivative of the function f(x) = (x^2 + 1) / (x - 1).

Accepted Answer

2x / (x - 1)^2 - (x^2 + 1) / (x - 1)^2

Answer

2x / (x - 1)

Answer

(x^2 + 1) / (x - 1)^2

Question 9

Which condition must be met for a function to be continuous at a point?

Accepted Answer

The left-hand limit and right-hand limit of the function at the point are equal.

Answer

The right-hand limit of the function at the point exists.

Question 10

Which statement best describes the relationship between the derivative of a function and the slope of its graph?

Accepted Answer

The derivative at a point equals the slope of the tangent line at that point.

Answer

The derivative at a point equals the slope of the secant line near that point.

Answer

The derivative at a point equals the average rate of change over a small interval.

Answer

The derivative at a point equals the slope of the normal line at that point.

Question 11

In the context of motion, what does the area under the curve of a velocity function over a time interval represent?

Accepted Answer

Distance traveled

Answer

Acceleration

Answer

Work done

Answer

Kinetic energy

Question 12

For a continuous and differentiable function f(x) on [a, b], how does the area under its curve relate to the definite integral of f(x) from a to b?

Accepted Answer

The area under the curve equals the definite integral.

Answer

The area under the curve is always positive, unlike the definite integral.

Answer

The area under the curve equals the average value of f(x) on [a, b] multiplied by the interval length.

Question 13

For the function f(x) = (x^2 - 4) / (x - 2), at which value of x is the function undefined?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 1

Answer

x = 3

Question 14

Is the function f(x) = (x^2 - 1) / (x - 1) continuous at x = 1?

Accepted Answer

Yes, because the limit of f(x) as x approaches 1 exists and equals f(1).

Answer

No, because the limit of f(x) as x approaches 1 does not exist.

Answer

Yes, because f(x) is defined at x = 1.

Answer

No, because the derivative of f(x) does not exist at x = 1.

Question 15

In the limit definition of the derivative, why is it crucial to take the limit as the increment (h) approaches zero?

Accepted Answer

Eliminating the finite increment's influence reveals the function's true rate of change, the derivative.

Answer

Determining the slope of the tangent line at a specific point is not the primary purpose of this limit.

Answer

While related, calculating the instantaneous rate of change is not the main focus of this limit.

Question 16

What is the derivative of a constant function?

Accepted Answer

Zero

Answer

A constant with the same value as the original function

Answer

The function itself

Answer

Undefined

Question 17

What does the area under the curve of a velocity-time graph represent?

Accepted Answer

Displacement

Answer

Acceleration

Answer

Kinetic Energy

Question 18

Determine the convergence of the improper integral:
 $$int_1^∞ \frac{dx}{x^2 + 1}$$
Select the correct answer from the options below:

Accepted Answer

The integral diverges.

Answer

The integral converges to 0.

Answer

The integral converges to 1.

Answer

The integral converges to π/2.

Question 19

Find the derivative of f(x) = x^3 - 3x^2 + 2x + 1. Which of the following is the first-degree term in f'(x)?

Accepted Answer

3x

Answer

0

Answer

2

Answer

x

Question 20

Determine the Maclaurin series representation of the function f(x) = sin(x) and specify its radius of convergence.

Accepted Answer

Maclaurin series: x - x^3/3! + x^5/5! - x^7/7! + ...; Radius of convergence: ∞

Answer

Maclaurin series: x + x^3/3! + x^5/5! + x^7/7! + ...; Radius of convergence: 1

Answer

Maclaurin series: sin(x); Radius of convergence: 0

Question 21

Calculate the derivative of the function f(x) = x^3 - 2x + 1 using the power rule.

Accepted Answer

3x^2 - 2

Answer

x^3 - 2x

Answer

x^2 - 2

Answer

3x - 2

Question 22

Calculate the area under the curve of the function f(x) = x^2 between the points x = 0 and x = 2 using definite integration.

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 23

State the Fundamental Theorem of Calculus, which establishes a connection between differentiation and integration.

Accepted Answer

The integral of a derivative is the original function.

Answer

The derivative of an integral is the original function.

Answer

The limit of a sum is the sum of the limits.

Answer

The area under a curve is equal to the integral of the function.

Question 24

Calculate the work done by a force F(x) = x^2 + 1 over a displacement of 3 units using integration.

Accepted Answer

12

Answer

4

Answer

10

Answer

14

Question 25

Which of the following improper integrals is convergent, meaning it has a finite value?

Accepted Answer

∫[1, ∞] 1/x dx

Answer

∫[0, ∞] x^2 dx

Answer

∫[-1, 1] 1/x^2 dx

Answer

∫[0, 1] 1/√x dx

Question 26

Find the Taylor series expansion of the function f(x) = cos(x) centered at x = π/2. This expansion allows us to approximate the function using a polynomial.

Accepted Answer

∑[n=0,∞] (-1)^n (x - π/2)^2n/(2n)!

Answer

∑[n=0,∞] (-1)^n (x - π/2)^2n/(2n + 1)!

Answer

∑[n=0,∞] (-1)^n (x - π/2)^n/n!

Answer

∑[n=0,∞] (-1)^n (x - π/2)^2n+1/n!

Question 27

Find the derivative of the function f(x) = x^3 + 4x^2 - 2.

Accepted Answer

3x^2 + 8x

Answer

x^2 + 4x - 2

Answer

3x^3 + 4x

Answer

x^3 + 4x + 2

Question 28

Identify the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

x

Answer

1/x^2

Answer

x^2

Question 29

Calculate the indefinite integral of ∫(x² + 3x - 5) dx .

Accepted Answer

x³/3 + 3x²/2 - 5x + C

Answer

3x³ + 9x²/2 - 15x + C

Answer

x²/2 + 3x - 5 + C

Answer

3x²/2 + 9x - 5 + C

Question 30

Find the equation of the tangent line to the curve y = x² - 2x at the point (1, -1).

Accepted Answer

y = x - 2

Answer

y = x + 2

Answer

y = 2x - 3

Answer

y = -2x + 1

Question 31

Determine the convergence or divergence of the series Σ(n=1 to ∞) (n² + 1) / (n³ + n).

Accepted Answer

Diverges

Answer

Converges

Answer

The test is inconclusive.

Answer

Cannot be determined

Question 32

Find the limit as x approaches 0 of the function f(x) = (sin x) / x.

Accepted Answer

1

Answer

0

Answer

∞

Answer

Does not exist

Question 33

Which of the following is NOT an application of calculus in mechanical engineering?

Accepted Answer

Time management

Answer

Fluid flow analysis

Answer

Heat transfer analysis

Answer

Stress and strain analysis

Question 34

Which of the following is the formula for the area under the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[x, y] f(x) dy

Answer

∫[y, x] f(x) dy

Question 35

Find the critical points of the function f(x) = x³ - 3x.

Accepted Answer

x = 0, ±1

Answer

x = 1

Answer

x = -1

Answer

x = 0

Question 36

Find the Maclaurin series representation of the function f(x) = e^x and determine its radius of convergence.

Accepted Answer

Σ (x^n / n!) from n = 0 to ∞

Answer

Σ (x^n / n) from n = 1 to ∞

Answer

Σ (x^n / (n + 1)) from n = 0 to ∞

Answer

Σ (x^n / 2^n) from n = 1 to ∞

Question 37

Find the critical points of the function f(x) = x³ + 2x² - 4x + 1. Use the first derivative test to determine the intervals where the function is increasing or decreasing.

Accepted Answer

Critical points: x = -1, 1; Increasing on (-∞, -1) and (1, ∞); Decreasing on (-1, 1)

Answer

Critical points: x = -1, 2; Increasing on (-∞, -1) and (2, ∞); Decreasing on (-1, 2)

Answer

Critical points: x = 1, 2; Increasing on (-∞, 1) and (2, ∞); Decreasing on (1, 2)

Answer

Critical points: x = -1, 0; Increasing on (-∞, -1) and (0, ∞); Decreasing on (-1, 0)

Question 38

Find the derivative of the function f(x) = x³ - 2x² + 5x - 1.

Accepted Answer

3x² - 4x + 5

Answer

x² - 4x + 5

Answer

3x - 4

Answer

x - 4

Question 39

Calculate the integral of the function f(x) = sin(x) from 0 to π/2.

Accepted Answer

1

Answer

0

Answer

2

Answer

π

Question 40

Determine the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Answer

Does not exist

Question 41

Determine if the function f(x) = x⁴ - 2x² + 1 is even, odd, or neither.

Accepted Answer

Even

Answer

Odd

Answer

Neither

Question 42

Determine the convergence or divergence of the improper integral ∫[1,∞] 1/x dx.

Accepted Answer

Diverges

Answer

Converges

Answer

Cannot be determined

Question 43

Consider the region bounded by the curves y = sin(x) and y = cos(x) between x = 0 and x = π/4. Find the volume of the solid generated when this region is rotated about the x-axis.

Accepted Answer

(B) π/2

Answer

(A) 1

Answer

(C) π

Question 44

Find the derivative of the function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 4x + 5

Answer

x² - 4x + 5

Question 45

Evaluate the definite integral of the function f(x) = sin(x) from x = 0 to x = π/2.

Accepted Answer

1

Answer

-1

Answer

0

Answer

π/2

Question 46

Determine if the function f(x) = e^x is continuous at x = 0.

Accepted Answer

Yes, because the left-hand limit, right-hand limit, and function value at x = 0 are all equal.

Answer

No, because the left-hand limit and right-hand limit do not exist.

Answer

No, because the left-hand limit and right-hand limit are equal, but the function value at x = 0 is undefined.

Answer

Yes, because the left-hand limit and right-hand limit are equal, even though the function value at x = 0 is undefined.

Question 47

Which of the following is NOT a property of differentiable functions?

Accepted Answer

Differentiable functions may have vertical tangents.

Answer

Differentiable functions are continuous.

Answer

For a differentiable function, the derivative exists at every point in its domain.

Answer

Differentiable functions may have inflection points.

Question 48

Which of the following is the chain rule formula for derivatives?

Accepted Answer

d/dx(f(g(x))) = f'(g(x))*g'(x)

Answer

d/dx(f(g(x))) = f'(x)*g(x)

Answer

d/dx(f(g(x))) = f'(x)/g(x)

Answer

d/dx(f(g(x))) = f(x)'g(x)

Question 49

Evaluate the integral ∫(2x + 1) dx over the real numbers.

Accepted Answer

x² + x + C, where C is an arbitrary constant

Answer

2x² - x + C

Answer

2x² + x + C

Answer

x² - x + C

Question 50

Calculate the derivative of the function f(x) = sin(2x), where x is a real number.

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(2x)

Answer

-2cos(2x)

Question 51

Evaluate the integral ∫cos(x) dx over the real numbers.

Accepted Answer

sin(x) + C, where C is an arbitrary constant

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

-cos(x) + C

Question 52

Find the maximum value of the function f(x) = -x² + 4x over the real numbers.

Accepted Answer

4

Answer

0

Answer

2

Answer

-2

Question 53

Calculate the area under the curve y = x² from x = 0 to x = 2, where x is a real number.

Accepted Answer

8/3

Answer

16/3

Answer

4/3

Answer

2

Question 54

Calculate the derivative of the function f(x) = x³ + 2x² - 5.

Accepted Answer

3x² + 4x

Answer

x³ + 4x - 5

Answer

3x³ + 2x

Answer

x² + 4x

Question 55

Calculate the indefinite integral of the function f(x) = 1/(x² + 1).

Accepted Answer

arctan(x) + C

Answer

1/x + C

Answer

ln(x² + 1) + C

Answer

tan(x) + C

Question 56

Find the derivative of the quotient function f(x) = (x^2 + 3x - 5)/(x - 2) using the quotient rule.

Accepted Answer

(2x + 3)/(x - 2)^2

Answer

(x + 3)/(x - 2)

Answer

(x^2 + 3x - 5)/((x - 2)^2)

Answer

None of the above

Question 57

Evaluate the indefinite integral of f(x) = cos(x) + e^x. Which integration technique is applied here?

Accepted Answer

sin(x) + e^x + C

Answer

-sin(x) + e^x + C

Answer

cos(x) - e^x + C

Answer

None of the above

Question 58

Calculate the area bounded by the curve y = x^2 + 2x between x = 0 and x = 2 using definite integration.

Accepted Answer

10/3

Answer

8

Answer

2

Question 59

Find the inverse of the linear function f(x) = 2x + 1.

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

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

Answer

f^-1(x) does not exist

Answer

f^-1(x) = 1/2x + 1

Question 60

Determine the derivative of the parametric equations x = t^2 and y = t^3.

Accepted Answer

dy/dx = 3t/2

Answer

dy/dx = 3t^2/2

Answer

dy/dx = 2t/3

Answer

dy/dx = t/2

Question 61

Find the derivative of the polynomial function f(x) = 3x^2 - 2x + 1 using the power rule.

Accepted Answer

6x - 2

Answer

3x - 2

Answer

9x - 2

Answer

6x^2 - 2

Question 62

Determine the slope of the tangent line to the curve y = x^3 at the point (1, 1).

Accepted Answer

3

Answer

1

Answer

0

Answer

6

Question 63

Calculate the area under the curve y = x^2 between x = 0 and x = 2.

Accepted Answer

8/3

Answer

2

Answer

4

Question 64

Determine the volume of the solid generated by rotating the region bounded by the curve y = √x, the x-axis, and the line x = 4 about the x-axis. Apply the method of cylindrical shells.

Accepted Answer

8π

Answer

16π

Answer

32π

Answer

4π