Math Class 11 Notes
Class 11th Math seems tough?
Go through 11th Math quickly.The second-order derivative measures how rate of change of a function is itself changing. It tells you how slope of function is changing as well as the concavity of the function and acceleration. In simple words, it is the derivative of the first derivative.
This is an important topic of class 12 continuity and differentiability chapter that explains what second derivatives are and what they explain. Once you have completed this and other topics, you can start NCERT excercise of Continuity and Differentiability.
Mathematically, if f(x) is the function, then its first derivative will be fβ²(x) and the second derivative will be fβ²β²(x). It tells you whether the graph of a function is curving upward or downward. IfΒ fβ²β²(x)>0, the function is concaving up and if fβ²β²(x)<0, then the function is concaving down.
Let us consider a function and find its second order derivative. Suppose is a function. Its first derivative will be fβ²(x) = . The second derivative in this case will be fβ²β²(x) = 6x. At x = 1, fβ²β²(1)=6, which means the function will concave up at x = 1.
Math Class 11 Notes
Class 11th Math seems tough?
Go through 11th Math quickly.Math Class 12 Notes
Need to complete Math before exams?
Revise 12th Math Notes.CBSE board covers questions based on the application of second-order derivative to understand how much does the student know what this term means. The second derivative has different applications in different fields, as mentioned below:
Β
The NEET exam or JEE Main exam will cover questions around the properties of the second order derivative. Second-order derivative has several important mathematical properties that analyze the behaviour of functions, curves and other real-world phenomena. Let us take a look at each one of them:
1. The second order derivative determines the concavity of the graph of a function. Concavity helps in visualizing how a curve bends and how it sketches graphs accurately.
2. The second derivative test classifies critical points (where f'(x) = 0) as local maxima, local minima or at neither point.Β
3. Using second order derivative of the position function, it is possible to determine acceleration in physics.Β
4. Second derivative is related to curvature of curve which indicates how sharp a curve will bend at a given point.
5. For linear functions, the second derivative will always be zero since the slope is constant and does not change.
6. Second order derivative is the foundation to understand higher-order derivatives.
7. In order for the second derivative to exist at a point, the first derivative must be differentiable at that point.
Β Class 12 CBSE Notes
Worried about the pending board syllabus?
Revise 12th Class Notes.11th CBSE Notes
Class 11th topics left before exams?
Revise 11th CBSE notes.Let S = Then the number of elements in the set isβ¦β¦β¦β¦β¦β¦.
(2, 1) (1, 2), (2, 2) each element has 3 choice.
(3, 2) (2, 3) (3, 1) (1, 3) (3, 3) each element has two choices.
total function = 3 Γ 3 Γ 2 Γ 2 Γ 2 = 72
Case I
None of the pre image have 3 as image, total functions = 2 Γ 2 Γ 1 Γ 1 Γ 1 = 4
Case II
None of the pre images have 2 as image then number of function = 25 = 32
Case III
None of the pre image have either 3 or 2 as image
Total function = 15 = 1
Total number of onto function
= 72 β 4 β 32 + 1 = 37
The first derivative represents slope of a function at a point, whereas the second order derivative describe the changes in slope over the independent variable in the graph.
Questions based on second order derivative will be asked in IIT JAM exam and IISER entrance exam. In the graph, the line labelled as slope is the first derivative of given function. Second derivative of given function corresponds to curvature or concavity of graph. When second-order derivative is positive, the graph of the function is concave upwards. On the other hand, if the second order derivative value is negative, the graph will be curved downwards.
Maths Continuity and Differentiability Exam