rentability of a function at a point sider the function f(a)= e the derivative function at a=0 by evaluating the following limit 7(0+A) - 7(0) We will show that the derivative function, f(a), is not defined at a=0 using the limit definition of the c TE: Enter an expression that contains only h into the first blank ITE: Enter the mit into the second blank. Type 'DNE if you believe that the limit does not exist. Ivative of a function at a point

Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?

If we need to calculate the derivative of a function at a point, there are two ways we can think about doing this. For example suppose f() = x – x, and we need to determine the value of f(4). One option is to just calculate the derivative at that point by plugging the point into the limit definition, like this: f(4 + h) – f(4) ( (4 + h)² – (4 + h)) - (4² – 4) f'(4) = lim (16 + 8h + h2 -4 – h) - (16 - 4) lim 7h + h2 = lim h(7+ h) lim h = lim h→0 h lim 7+h = 7. || h h h Another option is to calculate f'(x) in terms of x, and then plug in a value for x at the end, like this: f(x + h) – f(x) (2+ h) – (x + h)) - (2² – a) (22 + 2xh + h² – z – h) – (22 – 2) lim f'(x) = lim = lim 2xh + h2 - h h(2x + h - 1) = lim h 0 h h = lim = lim h h h that is to say, f'(x) = 2x – 1, and therefore f'(4) = 2(4) – 1 = 7. Notice that we get the answer f'(4) = 7 both ways. The second approach might look somewhat more complicated at first, but it turns out to be much more efficient if we ever need to know the…

why setting the derivative of a function equal to zero allows us to find the maximum or minimum values of the function. Give an example of how this could be useful in biology.

Suppose that the function y in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

2.) State one of the two formulas for finding the derivative of a function f(x) that uses limits. Use this formula to find the derivative of f(x) = 7x. Does your answer match what the power rule says the derivative should be? Why or why not?