Calculus: Differential Calculus: Solved Problem Set II
Calculus  Differential Calculus  Problem Set II  Outline of Contents: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 ALevel; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE/AIEEE Anyone else who needs this Tutorial as a reference!
In this tutorial, we'll pick up somewhat challenging problems based on the concepts of differential calculus introduced so far. We'll examine the continuity and derivability of more complicated functions. We'll differentiate more complex functions; using substitutions and transformations. We'll also pick up examples of chain/successive differentiation to find the nth derivatives.
Here's a quick look at the functions which will be inspected in this tutorial.
1. Show that the function f(x) is derivable for every value of x but the derivative is not continuous for x = 0.
f (x) = x^{2 }sin (1/x) when x = 0
0 when x = 0
2. Examine the continuity and derivability in the interval (−∞, ∞) for the following function f(x) =
1 in − ∞ < x < 0
1 + sin x in 0 ≤ x < π
2+ (x− π)^{2 }in π ≤ x < ∞
3. Find the n^{th} derivative of x / (x^{2} + a^{2}) This will be solved using a technique involving complex numbers.
4. Find the n^{th} derivatives of
(i) cos^{4} x
(ii)e^{ax} cos^{2} x sin x
These will be solved using compound and multiple angle trigonometric identities.
Complete tutorial with examples, solutions to problems :
