{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sheet5

# sheet5 - x = 0 of the function f x = tan(2 | x | x 6 The...

This preview shows page 1. Sign up to view the full content.

CM221A ANALYSIS I Exercise Sheet 5 Calculate the derivatives of each of the following two functions. Find all of the local and global maximum and minimum values, if there are any, of the functions on the intervals given. Sketch the graphs of the functions in these regions and explain your results! 1. f ( x ) = 1 1+log( x ) 2 on (0 , e]; 2. f ( x ) = xe - x 2 / 2 on [0 , ); 3. Evaluate the following derivative (you must state the conditions under which the result holds) d d x f ( x ) 2 g ( x ) 3 ! . 4. Evaluate the derivatives of the functions f ( x ) = x x for all real numbers x > 0. You will need to use the standard rules for differentiating the exponential and log functions. 5. Calculate the derivative from the left and derivative from the right at
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x = 0 of the function f ( x ) = tan(2 | x | + x ) . 6. The function f ( x ) is deﬁned for all x ∈ R by f ( x ) = ( x 2 (sin(1 /x ) if x 6 = 0 otherwise. Calculate the derivative of f ( x ) at all non-zero x . Using only the ε,δ deﬁ-nition of diﬀerentiation prove that f is diﬀerentiable at x = 0 and ﬁnd its derivative at that point. Sketch the graph of the derivative. Is the derivative continuous at x = 0? 7. Let g ( x ) = e-1 /x 2 if x 6 = 0 and g (0) = 0. Calculate the ﬁrst two derivatives of g ( x ) at x = 0. Explain your answer! 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online