Various problems

[Lippi86]

Having some problems with these problems

a) f(x) = ln (X^x). Find f'(x) and f''(x) . Hint: remember the ln?formulas...

b) f(x) = (x^2 - 4) / (x-2) and g(x) = x+2 both with their natural domains. Are these two functions exactly the same? If not, how do they differ? Your answer must include the concept “limit”.

c) Find the derivative of f(x) = xe^x + 1/x^2+2 + e where e = 2.718...

d) Find the largest and the smallest values of f(x) = x - ?x on the domain 0 < x < 9.

e) If f(x) = ax^2 + bx + c with a > 0 and c < 0 . Explain why the equation f(x) = 0 must
have two real solutions.


Thanks for help!! This is killing me...

 

[mmm4444bot]

e) If f(x) = ax^2 + bx + c with a > 0 and c < 0 . Explain why the equation f(x) = 0 must
have two real solutions.

With a > 0 and c < 0, what is the sign of the discriminant? It's positive. You can explain that, yes?

Next, what happens, in the Quadratic Formula, when the discriminant (D) is positive?

\(\displaystyle x = \frac{-b \pm \sqrt{D}}{2a}\)

We know what happens when D = 0: there is only one solution because \(\displaystyle \pm \sqrt{0}\) goes away.

We know what happens when D < 0: the solutions involve the imaginary unit i, so they are not Real.

 

[soroban]

Hello, Lippi86!

\(\displaystyle a) \;f(x) \:=\:\ln (x^x).\;\;\text{ Find }f'(x)\text{ and }f''(x)\)

\(\displaystyle \text{We have: }\;f(x) \;=\;x\cdot \ln(x)\)

\(\displaystyle \text{Then: }\;f'(x) \;=\;x\cdot\frac{1}{x} + 1\cdot\ln(x) \;=\;1 + \ln(x)\)

\(\displaystyle \text{And: }\;f''(x) \;=\;\frac{1}{x}\)

\(\displaystyle b)\;\;\begin{array}{ccc}f(x) &=& \frac{x^2 - 4}{x-2} \\g(x) &=& x+2\end{array}\;\text{ both with their natural domains.}\)

Are these two functions exactly the same? .If not, how do they differ?
Your answer must include the concept “limit”.

\(\displaystyle f(x) \:=\:\frac{(x-2)(x+2)}{x-2} \:=\:x+2\quad\text{ providing }x \neq 2\)

\(\displaystyle \text{Its graph looks like this:}\)

      |
      |           *
      |         *
      |       *
    4 + . . o (2,4)
      |   * ,
      | *   ,
      *     ,
    * |     ,
  - - + - - + - - - - -
      |     2
      |

There is a "hole" at (2,4) where the function does bnot exist.

\(\displaystyle \text{However: }\;\lim_{x\to2}\frac{(x-2)(x+2)}{x-2} \;=\;\lim_{x\to2}(x+2) \;=\;4\)



\(\displaystyle \text{The graph of }g(x) \:=\:x+2\,\text{ is the entire line.}\)

      |
      |           *
      |         *
      |       *
      |     *
      |   *
      | *
      *
    * |
  - - + - - - - - - - -
      |

 

[Lippi86]

Thank you very much, its a start!

 

[mmm4444bot]

b) f(x) = (x^2 - 4) / (x-2) and g(x) = x+2 both with their natural domains. Are these two functions exactly the same? If not, how do they differ? Your answer must include the concept “limit”.

HINT: If two functions are identical, then they both must have identical domains.

As far as using limits to describe how two functions differ, do you understand the concept of a limit?

c) Find the derivative of f(x) = xe^x + 1/x^2+2 + e where e = 2.718...

We can determine the first derivative of function f by differentiating term-by-term. In other words:

f '(x) = [x e^x] ' + [1/x^2 + 2] ' + [e] '

By the way, your typing 1/x^2+2 is not clear. (See note below.)

d) Find the largest and the smallest values of f(x) = x - ?x on the domain 0 < x < 9.

Use the first derivate of f '(x) to find local extrema of f. Any extrema within the interval (0, 9) are definitely candidates for largest and smallest value on this restricted domain.

However, be careful. Since the endpoints of the interval are excluded, both a largest and smallest value might not exist. In other words, there might be a largest value OR there might be a smallest value, but not necessarily both.

Do you understand why?

(I think that exercise (d) would be better written as, "Find any largest or smallest values ..." because this wording does not imply that both can be found within the given open interval.

Note: When we type algebraic ratios (that is, fractions with algebraic expressions for numerators and/or denominators), we must use grouping symbols.

For example, anytime a numerator or denominator is a sum of terms or a difference of terms, type parentheses around it. Otherwise, it's not clear what goes where (in the top or bottom or neither). Grouping symbols make it clear.

EGs:

Typing 1/x^2+2 actually means this: \(\displaystyle \frac{1}{x^2} \ + \ 2\)

Typing 1/(x^2 + 2) means this: \(\displaystyle \frac{1}{x^2 + 2}\)

"See" the difference? Which form is correct, for exercise (c) ?

 

[Lippi86]

thank you so long!

the one on the bottom is the correct function!

the one you´ve written last..

(1/(x^2+2))