2 Lecture

1. What is the absolute value of -9? a. -9 b. 9 c. 0 d. Undefined Answer: b. 9

   
   

2. What is the absolute value of 0? a. -1 b. 0 c. 1 d. Undefined Answer: b. 0

   
   

3. What is the derivative of the absolute value function? a. 1/x b. -1/x c. 0 d. step function Answer: d. step function

   
   

4. Which of the following is true about the absolute value function? a. It is a continuous function for all real numbers. b. It is a discontinuous function for all real numbers. c. It is a differentiable function for all real numbers. d. It is an odd function. Answer: a. It is a continuous function for all real numbers.

   
   

5. What is the range of the absolute value function? a. (-?, ?) b. [0, ?) c. [0, 1) d. [-1, 1] Answer: b. [0, ?)

   
   

6. Which of the following is true about the absolute value function graph? a. It is a straight line passing through the origin. b. It is a straight line passing through the point (1,1). c. It is a V-shaped curve with the vertex at the origin. d. It is a U-shaped curve with the vertex at the origin. Answer: c. It is a V-shaped curve with the vertex at the origin.

   
   

7. What is the limit of the absolute value function as x approaches infinity? a. -? b. ? c. 0 d. Does not exist Answer: b. ?

   
   

8. Which of the following is true about the absolute value of a negative number? a. It is negative. b. It is positive. c. It is zero. d. It is undefined. Answer: b. It is positive.

   
   

9. What is the distance between points (3,4) and (1,2)? a. 1 b. 2 c. ?2 d. ?10 Answer: c. ?2

   
   

10. Which of the following is true about the integral of the absolute value function? a. It is always positive. b. It is always negative. c. It is always zero. d. It can be positive, negative, or zero depending on the limits of integration. Answer: d. It can be positive, negative, or zero depending on the limits of integration.

1. What is the Absolute Value of -10? Answer: The Absolute Value of -10 is 10.

   
   

2. Define the Absolute Value function. Answer: The Absolute Value function is a function that returns the magnitude or distance of a number from zero on the number line, regardless of its sign. It is denoted by f(x) = |x|.

   
   

3. What is the graph of the Absolute Value function? Answer: The graph of the Absolute Value function is a V-shaped curve with its vertex at the origin.

   
   

4. Is the Absolute Value function continuous for all real numbers? Answer: Yes, the Absolute Value function is continuous for all real numbers.

   
   

5. What is the derivative of the Absolute Value function? Answer: The derivative of the Absolute Value function is a step function, which changes its value abruptly at x = 0. The derivative of the Absolute Value function is given by f’(x) = -1, for x < 0 and f’(x) = 1, for x > 0.

   
   

6. What is the limit of the function f(x) = |x| as x approaches 0? Answer: The limit of the function f(x) as x approaches 0 from the left is -0, and the limit of the function as x approaches 0 from the right is 0. Hence, the limit of the function f(x) as x approaches 0 does not exist.

   
   

7. Is the Absolute Value function differentiable at x = 0? Answer: No, the Absolute Value function is not differentiable at x = 0.

   
   

8. What is the distance between points (3, 4) and (-2, 1)? Answer: The distance between the points (3, 4) and (-2, 1) is given by |3 - (-2)| + |4 - 1| = 5 + 3 = 8.

   
   

9. How can we evaluate the integral ?[0, 2] |x - 1| dx? Answer: We can split the integral into two parts ?[0, 1] (1 - x) dx and ?[1, 2] (x - 1) dx, which gives the value of the integral as 1.

   
   

10. What is the value of |5 – 7| + |10 – 7|? Answer: The value of |5 – 7| + |10 – 7| is 2 + 3 = 5.

Calculus and Analytical Geometry are the branches of mathematics that deal with the study of functions, limits, derivatives, integrals, and geometrical concepts. In calculus, the concept of Absolute Value is crucial for solving various mathematical problems. Absolute Value, also known as the modulus, is a mathematical function that represents the distance of a number from zero, regardless of its sign. In this article, we will explore the concept of Absolute Value in Calculus and Analytical Geometry.

Definition of Absolute Value:

The Absolute Value of a number ‘a’ is denoted by |a|. The Absolute Value of a number is defined as the magnitude or the distance between that number and zero on the number line. The Absolute Value of a number is always non-negative, i.e., it is greater than or equal to zero. For instance, the Absolute Value of 5 is 5, and the Absolute Value of -5 is also 5.

Absolute Value Function:

The Absolute Value Function is a piecewise function that returns the Absolute Value of its input. The Absolute Value Function is denoted by f(x) = |x|. The graph of the Absolute Value Function is a V-shaped curve with its vertex at the origin. The Absolute Value Function is symmetric about the y-axis. The function has the following properties:

1. The Absolute Value Function is continuous for all real numbers.
2. The Absolute Value Function is differentiable for all x ? 0.
3. The derivative of the Absolute Value Function is discontinuous at x = 0.

Applications of Absolute Value in Calculus:

Absolute Value has numerous applications in calculus. Some of them are listed below:

1. Limit of a function: The limit of a function f(x) as x approaches a is defined as the value that f(x) approaches as x gets closer and closer to a. The Absolute Value function plays a crucial role in finding the limit of a function. For example, consider the function f(x) = |x|. The limit of the function as x approaches 0 from the left is -0, and the limit of the function as x approaches 0 from the right is 0. Hence, the limit of the function f(x) as x approaches 0 does not exist.
2. Derivatives: The Absolute Value function is not differentiable at x = 0. However, the function is differentiable for all x ? 0. The derivative of the Absolute Value function is a step function, which changes its value abruptly at x = 0. The derivative of the Absolute Value function is given by f’(x) = -1, for x < 0 and f’(x) = 1, for x > 0.
3. Integrals: The Absolute Value function is used in evaluating definite integrals. For example, consider the integral ?[0, 2] |x - 1| dx. We can split the integral into two parts ?[0, 1] (1 - x) dx and ?[1, 2] (x - 1) dx, which gives the value of the integral as 1.

Applications of Absolute Value in Analytical Geometry:

Absolute Value also has various applications in analytical geometry. Some of them are listed below:

1. Distance between two points: The Absolute Value function is used to find the distance between two points in a plane. Consider two points (x1, y1) and (x2, y2). The distance between the two points is given by |x2 - x1| + |y2 - y1|.
2. Graphs of equations: The Absolute Value function is used to graph equations. For instance, the graph of the equation |x| + |y| = 1