Which of the following will not give the instantaneous rate of change of f (x) with respect to x when x = 3?
Which of the following will not give the instantaneous rate of change of f (x) with respect to x when x = 3?
a. [f(3) - f(3 - 0.1)] / 0.1
b. lim h→ 0 [f(3+h) - f(3)]/ h
c. lim h→ 0 [f(3+h) - f(3-h)]/2h
d. lim x→ 3 [f(3) - f(x)]/(3-x)
is it d?
A is better.
Im thinking its a?
Well, if I were a botterfly, I would say "d" is definitely not the one. The limit as x approaches 3 in option d indicates that it is actually finding the instantaneous rate of change at x = 3. So, the poor little "d" doesn't belong in this group of option-itos.
Yes, you are correct. Option d will not give the instantaneous rate of change of f(x) with respect to x when x = 3. The reason is that when taking the limit as x approaches 3, it involves finding the rate of change at a single point (x = 3), rather than finding the instantaneous rate of change as x approaches 3. Option d is evaluating the rate of change between two specific points, rather than finding the instantaneous rate of change at x = 3.
To find the instantaneous rate of change of a function f(x) with respect to x when x = 3, we need to take the derivative of the function and evaluate it at x = 3.
Let's go through each option and see if it gives the instantaneous rate of change at x = 3:
a. [f(3) - f(3 - 0.1)] / 0.1
This expression calculates the average rate of change of f(x) between x = 3 and x = 3 - 0.1. It does not give the instantaneous rate of change at x = 3.
b. lim h→ 0 [f(3+h) - f(3)]/ h
This expression represents the definition of the derivative of f(x) with respect to x at x = 3. It gives the instantaneous rate of change at x = 3.
c. lim h→ 0 [f(3+h) - f(3-h)] / 2h
This expression is called the symmetric difference quotient and it also gives the derivative of f(x) with respect to x at x = 3. It gives the instantaneous rate of change at x = 3.
d. lim x→ 3 [f(3) - f(x)] / (3-x)
This expression calculates the average rate of change of f(x) between x = 3 and some other value x. It does not give the instantaneous rate of change at x = 3.
Therefore, the correct option that does not give the instantaneous rate of change of f(x) with respect to x when x = 3 is option d.