Sorry for posting another quesiton..;
f(x) = { -7x^2 + 7x for x<=0
{ 3x^2 + 7x for x>0
According to the definition of derivative, to compute f'(0), we need to compute the left-hand limit
lim x-> 0− = _____
and the right-hand limit
lim x->0+ ______
We conclude that f'(0)= _____
Write DNE if the derivative does not exist.
Ok... I know that the answer for f'(0) is 7... I thought that the answers for the previous 2 were 0 because I was left with -7x for the left one and 3x for the right one... what am I doing wrong?
For x-> 0-,
f'(x) = d(-7x^2 + 7x)/dx = -14x+7 = 7
For x-> 0+,
f'(x) = d(3x^2 + 7x)/dx = 6x + 7 = 7
Since f'(0-) = f'(0+) = 7,
f'(0) exists, and equal to 7.
I already tried 7 for the first two... but the web ways that the answer is incorrect...
I have no idea why that's not the answer.....
You may want to check if there is no typo somewhere.
Other than that, I cannot think of anything else.
To compute the left-hand limit, we need to evaluate the expression for f(x) as x approaches 0 from the negative side (i.e., values of x less than 0). In this case, the expression for f(x) is -7x^2 + 7x.
To compute the right-hand limit, we evaluate the expression for f(x) as x approaches 0 from the positive side (i.e., values of x greater than 0). In this case, the expression for f(x) is 3x^2 + 7x.
To compute the left-hand limit, we substitute the value of x as a small negative number, say -0.01. Thus, we have:
lim x->0- (-7x^2 + 7x)
= -7(-0.01)^2 + 7(-0.01)
= -7(0.0001) - 0.07
= -0.0007 - 0.07
= -0.0707
Similarly, to compute the right-hand limit, we substitute the value of x as a small positive number, say 0.01:
lim x->0+ (3x^2 + 7x)
= 3(0.01)^2 + 7(0.01)
= 3(0.0001) + 0.07
= 0.0003 + 0.07
= 0.0703
Therefore, by evaluating the left-hand limit and the right-hand limit, we find that f'(0) does not exist because the left-hand limit is -0.0707 and the right-hand limit is 0.0703, which are not equal.
It seems that there was a misunderstanding in your previous calculation. The terms -7x and 3x were not the only terms in the expressions for the left-hand and right-hand limits of f(x). The terms also involve x^2.