Question

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) = e^rx Determine the values of r so that f satisfies the equation f"(x) + f'(x) - 6f(x) = 0.

Answer 1

1. The function $f(x)$ is called odd if and only if $-f(x)=f(-x).$ It is called even if and only if $f(x)=-f(x)$.

(a) The function is odd, since $-x^3=(-x)^3$ .

(b) The function is even, since $cos(x)=cos(-x)$ .

2. The mean value Theorem states the following: Suppose that function $f(x)$ is continuous on the closed interval $[a,b]$ and is differentiable on the open interval $(a,b)$. Then there exists the point $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.

3. (a) The derivative of the function is: $y'=4x-24x^{-3}$. After substituting $x=2$ we receive: $y'(2)=8-\frac{24}{8}=8-3=5$.

(b) We receive: $f(x)+7=-\frac3x$. It yields: $x=-\frac{3}{f(x)+7}$. It can be also rewritten as: $f^{-1}(x)=-\frac{3}{x+7}$.

4. We substitute $f(x)=e^{rx}$ and receive: $r^2+r-6=0$ ; $r=\frac{-1\pm5}{2}=2;-3.$ The solutions are: $f(x)=e^{2x}$ and $f(x)=e^{-3x}$.