Use the alternate definition, lim x->a f(x)-f(a)/x-a, to determine the instantaneous rate of change of f(x)= 1/x at the point (1,1/5)
a)1/5
b)-1/5
c)1/25
d)-1/25
To determine the instantaneous rate of change of a function using the alternate definition of the limit, we need to find the value of the limit:
lim x->a (f(x) - f(a)) / (x - a)
In this case, we have the function f(x) = 1/x and the point (1, 1/5) with a = 1.
So, substituting the values into the formula, we get:
lim x->1 (1/x - 1/1) / (x - 1)
Simplifying this expression, we have:
lim x->1 (1/x - 1) / (x - 1)
Now, let's substitute x = 1 and simplify:
(1/1 - 1) / (1 - 1) = (1 - 1) / 0
We have an indeterminate form of 0/0, which means we need to apply L'Hôpital's Rule. Differentiating the top and bottom separately, we get:
lim x->1 (-1/x^2) / 1
Now, let's substitute x = 1:
(-1/1^2) / 1 = -1/1 = -1
Therefore, the instantaneous rate of change of f(x) = 1/x at the point (1, 1/5) is -1.
Therefore, the correct option is option b) -1/5.
To determine the instantaneous rate of change of a function using the alternate definition of the limit, we need to evaluate the expression lim(x->a) [f(x) - f(a)] / (x - a), where f(x) is the given function, and (a, f(a)) is the point of interest.
In this case, we have f(x) = 1/x, and the point of interest is (1, 1/5). We need to find the limit as x approaches 1 of [f(x) - f(1)] / (x - 1).
Step 1: Evaluate f(1)
To find f(1), substitute x = 1 into the function f(x) = 1/x:
f(1) = 1/1 = 1
Step 2: Set up the expression for the limit
Substitute the values we have into the expression for the limit:
lim(x->1) [f(x) - f(1)] / (x - 1) = lim(x->1) [(1/x) - 1] / (x - 1)
Step 3: Simplify the expression
To simplify this expression, we need to find a common denominator. Multiply the numerator and denominator by x to get rid of the fractions:
lim(x->1) [(1 - x) / (x * (x - 1))]
Step 4: Cancel out common factors
Simplify further by cancelling out the common factors (x - 1) in the numerator and denominator:
lim(x->1) [-1 / (x * 1)] = lim(x->1) -1/x
Step 5: Evaluate the limit
To find the limit as x approaches 1 of -1/x, substitute x = 1 into the expression:
lim(x->1) -1/x = -1/1 = -1
Therefore, the instantaneous rate of change of f(x) = 1/x at the point (1, 1/5) is -1.
So, the correct answer is:
b) -1/5
just apply the definition. What's the problem?
f(x) = 1/x
f(a) = 1/a
f(x)-f(a) = (a-x)/(ax)
Now divide that by x-a and you have -1/(ax)
the limit as x->a is -1/x^2
So, fix your typo and adjust the evaluation.
(1,1/5) is not on the curve for 1/x.