I. Investigate whether the following functions are even or odd:
(a) f(x) = x3
(b) f(x) = cos x
II. State the mean value theorem
III. (a) Find the derivative of the function y = 2x2 + 12/x2 when x = 2
(b) f(x) = -3/x-7. Find the inverse of the function.
Iv. Consider the function f(x) = erx Determine the values of r so that f satisfies the equation f"(x) + f'(x) - 6f(x) = 0.
I)
A function f(x) is said to be:
even, if f(-x) =f(x) and
odd, if f(-x)=-f(x)
Therefore, for
(a) f(x)=x3 , we have f(-x)=(-x)3 = (-1)3(x)3= -x3=-f(x)
Hence, f(x)=x3 is an odd function.
(b) f(x)=cos x,
so, f(-x)=cos (-x)=cos (0-x)
Using the identity, cos (A-B)=cos A cos B + sin A sin B, we have
f(-x)=cos (-x)=cos (0-x)=cos (0) cos (x) + sin (0) sin (x)
=cos (x) - 0 sin (x)
=cos (x)
=f(x)
Hence, cos (x) is an even function.
II) The mean value theorem is stated as follows:
Suppose g(x) is a function that is continuous on a closed interval [a, b] and
is also differentiable on the open interval (a, b) then there exists a number c
such that a<c<b and
III) a)
when x=2, we have
b) Let y=f(x)
So,
swap x with y:
solve for y:
Thus,
IV) From we have and
Substituting the derivatives into the equation we have
Since we have
Upon solving the quadratic equation, we have
So, or
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