The Learning Point‎ > ‎Mathematics‎ > ‎

Calculus - Functions, Limits, Continuity - Problem Set I - domain, range, plots , graphs; Invertibility; L'Hospital's rule


Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level;
Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule.

 Differential Equations




Outline of the Tutorial

Domain and Range


Finding the domains of the following functions.
(i) y = √(x 3 − x)
(ii) y = √(1 + 2sinx)
(iii) (x+3)/ √(x −5x+4)
(iv) y = sin−1 (log2 ( x2/2 ))

Drawing the graphs of the following functions.
(i) y = |x| + |x − 1|
(ii) y = [x], where [x] denotes the greatest integer not greater than x

Finding the domain and range of the function y = ln  (3x2 − 4x + 5).

Finding out if functions are invertible : Let f : R → R be defined by f (x) = (ex −e -x)/2 , Is f (x) invertible? If so, find its inverse.

Evaluating Limits:
lim (x2 − 16)/( x − 4)  as x->4
lim(x2 + x − 6) / (x2 - 4) as x->2
 
Evaluating the limits, as x-> 0, for : sin kx / x, tan x/ x ,  ( 1 - cos x)/ x , sin Ax / cos Bx and the limit for x ln x ( for x > 0)
Limits involving polynomials of 'x' in the numerator and denominator; limits involving series expansions, exponential and logarithmic functions.
A general familiarity with 0/0 and Infinity/Infinity kinds of limits. 

... apart from this, we will also find the limits for a lot more interesting functions and we will also learn to apply L'Hospital's rule.

Complete Tutorial with examples, solutions, problems and plots of functions (Also try out the MCQ Quiz at the end):



MCQ Quiz on Functions, Limits, Continuity- 1

Companion MCQ Quiz for Functions, Limits, Continuity #1 - test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz- Functions, Limits, Continuity - Part 1






Other Calculus Tutorials                          

Quick and introductory definitions related to Funtions, Limits and Continuity

Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions; L'Hospital's Rule

Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x  

Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems

Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc

Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions 

Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives -

Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values, Rolle's Theorem

Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching

Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves 

Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  

Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves  

Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios  

Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI 

Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations  

Integral Calculus - Solved Problems Set V- Integration as a summation of a series 

Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor 

Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution 

Differential Equations - Solved Problems - Set III - More Differential Equations  

Differential Equations - Solved Problems - Set IV 


    

 





Comments