# 2 Lecture

## MTH101

### Midterm & Final Term Short Notes

## Absolute Value

Absolute Value, also known as the modulus, is a mathematical function that represents the distance of a number from zero, regardless of its sign.

**Important Mcq's**

Midterm & Finalterm Prepration

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**What is the absolute value of -9?**a. -9 b. 9 c. 0 d. Undefined**Answer: b. 9****What is the absolute value of 0?**a. -1 b. 0 c. 1 d. Undefined**Answer: b. 0****What is the derivative of the absolute value function?**a. 1/x b. -1/x c. 0 d. step function**Answer: d. step function****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.****What is the range of the absolute value function?**a. (-?, ?) b. [0, ?) c. [0, 1) d. [-1, 1]**Answer: b. [0, ?)****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.****What is the limit of the absolute value function as x approaches infinity?**a. -? b. ? c. 0 d. Does not exist**Answer: b. ?****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.****What is the distance between points (3,4) and (1,2)?**a. 1 b. 2 c. ?2 d. ?10**Answer: c. ?2****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.**

**Subjective Short Notes**

Midterm & Finalterm Prepration

Past papers included

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**What is the Absolute Value of -10? Answer:**The Absolute Value of -10 is 10.**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|.**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.**Is the Absolute Value function continuous for all real numbers? Answer:**Yes, the Absolute Value function is continuous for all real numbers.**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.**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.**Is the Absolute Value function differentiable at x = 0? Answer:**No, the Absolute Value function is not differentiable at x = 0.**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.**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.**What is the value of |5 – 7| + |10 – 7|? Answer:**The value of |5 – 7| + |10 – 7| is 2 + 3 = 5.

### 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:- The Absolute Value Function is continuous for all real numbers.
- The Absolute Value Function is differentiable for all x ? 0.
- 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:

- 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.
- 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.
- 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:

- 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|.
- Graphs of equations: The Absolute Value function is used to graph equations. For instance, the graph of the equation |x| + |y| = 1