Course outline
Module 1
Module 1: Introduction to Functions and Graphs
Introduction to Functions and Graphs
Knowledge check
Q1. What is the domain of the function f(x) = 1/x?
- A. All real numbers
- B. All real numbers except 0 ✓
- C. All positive real numbers
- D. All negative real numbers
Why: The function is undefined at x = 0, so the domain excludes 0.
Q2. Which of the following is a function?
- A. A circle
- B. A parabola ✓
- C. A rectangle
- D. A triangle
Why: A parabola passes the vertical line test, meaning it is a function.
Q3. What is the range of the function f(x) = x²?
- A. All real numbers
- B. All non-negative real numbers ✓
- C. All positive real numbers
- D. All negative real numbers
Why: The square of any real number is non-negative, so the range is all non-negative real numbers.
Q4. Which of the following is the graph of a linear function?
- A. A circle
- B. A straight line ✓
- C. A parabola
- D. A hyperbola
Why: A linear function is represented by a straight line on the graph.
Q5. What is the value of f(2) if f(x) = 3x + 1?
Why: Substituting x = 2 into the function gives f(2) = 3(2) + 1 = 7.
Module 2
Module 2: Limits and Continuity
Limits and Continuity
Knowledge check
Q1. What is the definition of a limit?
- A. The value of the function at a specific point
- B. The value that the function approaches as the input approaches some value ✓
- C. The rate of change of the function
- D. The slope of the tangent line
Why: A limit describes the behavior of a function as the input approaches a certain value, not necessarily the value of the function at that point.
Q2. What does it mean for a function to be continuous at a point?
- A. The function has a vertical asymptote at that point
- B. The function is defined at that point and the limit as x approaches that point exists and equals the function value ✓
- C. The function has a removable discontinuity at that point
- D. The function oscillates infinitely near that point
Why: Continuity at a point requires the function to be defined there, the limit to exist, and the limit to equal the function value.
Q3. What is the limit of f(x) = 5 as x approaches 2?
- A. 0
- B. 5 ✓
- C. 2
- D. Undefined
Why: The function f(x) = 5 is a constant function, so its limit as x approaches any value is 5.
Q4. If the left-hand limit and right-hand limit of a function at a point are not equal, what can we say about the limit at that point?
- A. It exists and is equal to the left-hand limit
- B. It exists and is equal to the right-hand limit
- C. It does not exist ✓
- D. It is undefined
Why: For a limit to exist at a point, the left-hand and right-hand limits must be equal.
Q5. What is the limit of f(x) = (x^2 - 1)/(x - 1) as x approaches 1?
- A. 0
- B. 1
- C. 2 ✓
- D. Undefined
Why: Simplifying the expression gives f(x) = x + 1, and the limit as x approaches 1 is 2.
Module 3
Module 3: Derivatives and Rates of Change
Derivatives and Rates of Change
Knowledge check
Q1. What is the derivative of f(x) = 5x^2 at x = 2?
- A. 10
- B. 20 ✓
- C. 30
- D. 40
Why: The derivative of 5x^2 is 10x. Plugging in x = 2 gives 10*2 = 20.
Q2. What does the derivative represent?
- A. The slope of the tangent line ✓
- B. The area under the curve
- C. The maximum value of the function
- D. The rate of change of the function
Why: The derivative represents the slope of the tangent line to the function at a given point.
Q3. Which of the following is the derivative of f(x) = sin(x)?
- A. cos(x) ✓
- B. -sin(x)
- C. tan(x)
- D. cot(x)
Why: The derivative of sin(x) is cos(x).
Q4. What is the derivative of f(x) = e^x?
- A. e^x ✓
- B. x*e^(x-1)
- C. 0
- D. 1
Why: The derivative of e^x is e^x, which is a unique property of the exponential function.
Q5. What is the derivative of f(x) = 3x + 4?
Why: The derivative of 3x + 4 is 3, since the derivative of a constant is zero.
Module 4
Module 4: Applications of Derivatives
Applications of Derivatives
Knowledge check
Q1. What is the derivative of a function at a particular point called?
- A. The slope of the function
- B. The instantaneous rate of change ✓
- C. The average rate of change
- D. The limit of the function
Why: The derivative at a point represents the instantaneous rate of change of the function at that point.
Q2. If the derivative of a function is positive, what does that indicate about the function?
- A. The function is decreasing
- B. The function is increasing ✓
- C. The function is constant
- D. The function is undefined
Why: A positive derivative indicates that the function is increasing at that point.
Q3. What does the second derivative of a function tell us?
- A. The slope of the function
- B. The concavity of the function ✓
- C. The maximum value of the function
- D. The minimum value of the function
Why: The second derivative tells us about the concavity of the function.
Q4. What is a critical point of a function?
- A. A point where the function is undefined
- B. A point where the derivative is zero or undefined ✓
- C. A point where the function is maximum
- D. A point where the function is minimum
Why: A critical point occurs where the derivative is zero or undefined, and it is used to find local extrema.
Q5. What is the purpose of using the first derivative test?
- A. To find the concavity of the function
- B. To find the maximum and minimum values of the function ✓
- C. To find the inflection points of the function
- D. To find the domain of the function
Why: The first derivative test is used to determine the local maximum and minimum values of a function.
Module 5
Module 5: Integration and Accumulation
Integration and Accumulation
Knowledge check
Q1. What is the integral of 2x with respect to x?
- A. x^2 + C ✓
- B. x^2 + C
- C. 2x + C
- D. x^3/3 + C
Why: The integral of 2x is x^2 + C, since the derivative of x^2 is 2x.
Q2. What does the definite integral from a to b of f(x) dx represent?
- A. The derivative of f(x)
- B. The area under the curve of f(x) between a and b ✓
- C. The slope of the tangent line to f(x)
- D. The maximum value of f(x) on [a, b]
Why: The definite integral represents the net area under the curve of f(x) between the limits a and b.
Q3. What is the integral of 1/x with respect to x?
- A. x^2/2 + C
- B. ln|x| + C ✓
- C. e^x + C
- D. x + C
Why: The integral of 1/x is ln|x| + C, as the derivative of ln|x| is 1/x.
Q4. What is the integral of a constant, say 5, with respect to x?
- A. 5x + C ✓
- B. x^2 + C
- C. 5 + C
- D. 0 + C
Why: The integral of a constant k with respect to x is kx + C, so the integral of 5 is 5x + C.
Q5. What is the value of the definite integral from 0 to 2 of 3x^2 dx?
Why: The antiderivative of 3x^2 is x^3. Evaluating from 0 to 2 gives 2^3 - 0^3 = 8.
Module 6
Module 6: Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
Knowledge check
Q1. What does the Fundamental Theorem of Calculus connect?
- A. Differentiation and Integration ✓
- B. Algebra and Geometry
- C. Trigonometry and Calculus
- D. Limits and Series
Why: The Fundamental Theorem of Calculus connects differentiation and integration, showing they are inverse processes.
Q2. Who is credited with formalizing the Fundamental Theorem of Calculus?
- A. Isaac Newton
- B. Gottfried Wilhelm Leibniz
- C. Both Isaac Newton and Gottfried Wilhelm Leibniz ✓
- D. Leonhard Euler
Why: Both Isaac Newton and Gottfried Wilhelm Leibniz are credited with formalizing the Fundamental Theorem of Calculus, though independently.
Q3. According to the Fundamental Theorem of Calculus, if F(x) is an antiderivative of f(x), then what is the integral of f(x) from a to b?
- A. F(a) - F(b)
- B. F(b) - F(a) ✓
- C. F(a) + F(b)
- D. F(b) * F(a)
Why: According to the Fundamental Theorem of Calculus, the integral of f(x) from a to b is F(b) - F(a), where F(x) is an antiderivative of f(x).
Q4. What is the derivative of the integral from a to x of f(t) dt?
- A. f(x) ✓
- B. f(a)
- C. f(t)
- D. 0
Why: The derivative of the integral from a to x of f(t) dt is f(x), as stated by the first part of the Fundamental Theorem of Calculus.
Q5. What does the second part of the Fundamental Theorem of Calculus allow us to do?
- A. Compute definite integrals using antiderivatives ✓
- B. Compute indefinite integrals using derivatives
- C. Find the area under a curve using limits
- D. Find the slope of a tangent line using integrals
Why: The second part of the Fundamental Theorem of Calculus allows us to compute definite integrals using antiderivatives.
Module 7
Module 7: Basic Differential Equations
Basic Differential Equations
Knowledge check
Q1. What is the general solution to the differential equation dy/dx = 3x²?
- A. y = x³ + C ✓
- B. y = x³ + C
- C. y = 3x³ + C
- D. y = x² + C
Why: The integral of 3x² with respect to x is x³ + C, where C is the constant of integration.
Q2. Which of the following is a first-order differential equation?
- A. d²y/dx² + 2y = 0
- B. dy/dx + y = 5 ✓
- C. y'' + y' + y = 0
- D. y³ + x² = 10
Why: A first-order differential equation contains only the first derivative of the dependent variable.
Q3. What is the order of the differential equation d³y/dx³ + 2y = sin(x)?
Why: The order of a differential equation is determined by the highest derivative present in the equation.
Q4. What is the integrating factor for the linear differential equation dy/dx + 2y = x?
- A. e^(2x) ✓
- B. e^(-2x)
- C. e^(x)
- D. e^(-x)
Why: The integrating factor for dy/dx + P(x)y = Q(x) is e^(∫P(x)dx), which in this case is e^(2x).
Q5. Which of the following is a separable differential equation?
- A. dy/dx = x + y
- B. dy/dx = x/y ✓
- C. dy/dx + y = x²
- D. dy/dx = y + sin(x)
Why: A separable differential equation can be written as dy/dx = f(x)g(y), which is the case for dy/dx = x/y.