RD Sharma Class 12 Chapter 14 Solutions (Differentials Errors And Approximations)
RD Sharma Class 12 Maths Solutions Chapter 14 Differentials Errors and Approximations is designed for students like you who are aspiring to get good marks in CBSE Board exams and qualify in advanced engineering examinations such as IIT JEE Mains. The chapter of RD Sharma Class 12 Solutions teaches you about the basics of Differential, Errors and Approximations. This chapter talks about differentials, absolute error, relative error, percentage error, the geometrical meaning of differentials, algorithms, etc. This chapter gives you an insight on how to use differential errors and approximations to other concepts of Class 12th Mathematics.
The Chapter 14 of RD Sharma Class 12 Maths i.e. Differential, Errors and Approximations comprise 3 exercises and 61 questions testing your knowledge of the previous chapter combined with Differential, Errors and Approximations of Class 12 Mathematics. The solutions here are drafted for you in the easiest terminologies for quick grasp. You will find the questions are arranged in the same format as in RD Sharma Class 12 Maths Chapter 14 Differential Errors and Approximations.
Read and practise solutions from RD Sharma Class 12 Maths to gain knowledge of concepts and learn ways to solve a variety of questions. These chapters will prepare you the way you need to appear in CBSE exams and increase your opportunities for graduate-level studies.
Topics Covered in RD Sharma Class 12 Maths Solutions Chapter 14 Differential Errors and Approximations
Derivative of y w.r.t x is dy/dx as the limit of Δy/Δx as Δx approaches 0.
Consider dy/dx as a symbol.
Let y be a function having a value, y = f(x). Y is a function of x and Δx is considered a small change in the value of x.
Lim Δx ->0 (Δy/Δx) = dy/dx = f’(x)
=> ∆y/∆x = f’(x) + ∈, where ∈-> 0 as ∆x -> 0
=> ∆y = f’(x)∆x + ∈∆x
=> ∆y = f’(x)∆x, approximately
=> ∆y = dy/dx ∆x, approximately
You may find this formula very useful in calculating small changes or in other words, errors in a dependent variable corresponding to minor changes in an independent variable.
Some important terms
Relative Error – Consider ∆x an error in x, then (∆x/x) will be known as a relative error in x.
Absolute Error – ∆x in x will be called as absolute error in x
Percentage Error – Let ∆x be a small change in x, then (∆x/x)*100 is known as percentage error in x.
Geometrical Meaning of Differential
Let’s consider a point P(x,y) present on the curve y= f(x).
Where, F(x) = differentiable real function
Consider Q(x+∆x, y+∆y) the neighbouring points on the same curve I.e. y = f(x).
∆x = small change in the value of x
∆y = small change in the value of y
On the graphical representation, you’ll see ∆y/∆x = slope of secant PQ
However, as ∆x approaches 0 I.e. ∆x -> 0 :
∆y/∆x will approach dy/dx’s limiting value.
- Algorithm to find ∆y, an approximate change in y corresponding to ∆x, approximate change in x.
- Select an independent variable and choose its initial value. Let the independent variable by x and change in value be x+∆x.
- ∆x is a small change in x. Assume dx = ∆x
- Find the value of dy/dx from y relation which means y = f(x)
- Find dy/dx at points (x,y)
- Find the value of dy by using the dy relation = (dy/dx)dx
- Put ∆y ≈ dy as an y’s approximate change.
Discussion of Exercises of RD Sharma Class 12 Maths Chapter 14 Differentials, Errors and Approximations
This chapter contains only 3 exercises comprising 61 questions in total. You will need to have knowledge of previous chapters as well in order to apply methods and rules to Differentials, Errors, and Approximations. These solutions will give you an idea of the type of questions being asked in your upcoming CBSE and important entrance exams. The questions asked in the exercise of RD Sharma Class 12 Maths Chapter 14 Differentials, Errors and Approximations will ask you to find the approximate change in a given variable, approximate decrease or increase in the surface area or volume of an object with a specific shape, and finding the percentage error.
Benefits of RD Sharma Class 12 Maths Chapter 14 Differentials, Errors and Approximations
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