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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 2: Derivative as slope of tangent line

# Interpreting derivative challenge

Given that f(-2)=3 and f'(x)≤7, Sal finds the largest possible value of f(10). Created by Sal Khan.

## Want to join the conversation?

• I don't quite see how mean value theorem is related to this question...
• f(b)-f(a)/b-a = f ' (x), which is what was used to find out f(10) at the max f ' (x) of 7
• Is it correct if I solve it as following:
First, since the maximum slope of the function is 7, the largest number in the term of the function must increase with 7 to reach its max y. Thus, saying that the distance is |10 -(-2)| = 12, I could multiply 12 by, the scalar slope, 7; `12*7=84`. Well, because f(-2) is not zero but 3, I must increase 84 with 3. So, f(10) maximum y value is 87.

Am I fine solving it as above?
• Ye, I did the same too! But with a silly mistake i didn't see we start from -2 xD and i made it 7*10 +3 = 73 and then when i looked at your 12.. I got it :O it starts from -2 and not 0
(1 vote)
• At , Sal says that "the fastest growing function would be a line." Why couldn't some type of curve also be the fastest growing function? Would it be possible for such a curve to have 7 as the greatest instantaneous speed?
• The problem said that the f'(x) ≤ 7. Thus, the maximum slope is 7. Thus, the function that meets that requirement and increases at the maximum allowed rate would have to have a constant slope of 7 (the maximum slope allowed here). Only straight lines have a slope that is constant.
• According to this scenario, am I safe when assuming the maximum value of `x` which you have to find is always the latter endpoint?
• No, that is not always the case. You must check f(x) (not the derivative) for the two endpoints, as well as the local extrema within the interval.
• What is the mean value theorem?
• (at about ) Why is it f(10) - 3 for y? I'm wondering what the logic is behind putting the f(10) there.
(1 vote)
• We're trying to find the highest possible value of f(x) when x=10, as stated in the initial problem. Since we have a known point for the function and a bound on the derivative (slope), we can use the highest possible derivative to extrapolate from the known point f(-2)=3 to find the greatest possible value for f(10).
• at what is the mean value theorem??
(1 vote)
• Where did you get 84 from?
(1 vote)
• In the video at the formula for the slope is presented:
` ( f(10)-3 ) / ( 10 - - 2 ) = 7 ` .
This simplifies to: ` ( f(10)-3 ) / ( 12 ) = 7 ` . Then multiply both sides of the equation with 12. This results in: ` ( f(10)-3 ) = 7 * 12 = 84 `.
So ` f(10) = 84 + 3 = 87 ` !
(1 vote)
• Is there any kind of function that cannot be differentiated ?
(1 vote)
• Any function that is not continuous cannot be differentiated. Also, if there are any sudden changes in slope (such as a corner), you can't differentiate over the entire interval. You can get around this problem a bit by taking derivatives of different sections of the function's domain.
(1 vote)
• a competitive firm has the following production function y=f(x)=400x+60x2+6x3 where y=output,x=input.the firm faces an output and input prices of p=10 and an input prices of w=5440. 1-write a profit function of this firm in term of output and input prices and the input level. 2-what is the profit maximazing level of input for this firms?verify that the input level you choose is the profit maximizing points. 3-find the marginal product (MPx) of the variable input.
4-verify that P(MPx)=W at the profit maximizing input level.
(1 vote)

## Video transcript

Let f be a differentiable function for all x. If f of negative 2 is equal to 3 and f prime of x is less than or equal to 7 for all x, then what is the largest possible value of f of 10? And so I encourage you to think about this on your own, pause the video, try to figure out the largest possible value for f of 10. And then we'll work through it together. So I'm assuming you've given a go at it. So let's visualize this. So let me draw some axes here. So let's say that's my x-axis. That's my x-axis right over there. And this right over here is my y-axis. That's my y-axis [INAUDIBLE] I'll graph y equals f of x. And they tell us f of negative 2 is equal to 3. And the two axes aren't going to be drawn to scale. So let's say this is negative 2. And this right over here is the point negative 2 comma 3. And they tell us that f prime of x is a less than or equal to 7, that the instantaneous slope is always less than or equal to 7. So really, the way to get the largest possible value of f-- we don't have to necessarily invoke the mean value theorem, although the mean value theorem will help us know for sure-- is to say well, look, the largest possible value of f of 10 is essentially if we max this thing out. If we assume that the instantaneous rate of change just stays at the ceiling right at 7. So if we assumed that our function, the fastest growing function here would be a line that has a slope exactly equal to 7. So the slope of 7 would look-- and obviously, I'm not drawing this to scale. Visually, this looks more like a slope of 1, but we'll just assume this is a slope of 7 because it's not at the same-- the x and y are not at the same scale. So slope is equal to 7. And so if our slope is equal to 7, where do we get to when x is equal to 10? When x is equal to 10, which is right over here, well what's our change in x? So what's our change in x? Let's just think about it this way. Our change in y over change in x is going to be what? Well our change in y is going to be f of 10 minus f of 2. f of 2 is 3, so minus 3, over our change in x. Our change in x is 10 minus negative 2. 10 minus negative 2 is going to be equal to 7. This is the way to max out what our value of f of 10 might be. If at any point the slope were anything less than that, because remember, the instantaneous rate of change can never be more than that. So if we start off even a little bit lower, than the best we can do is get to that. Remember, we can't do something like that. That would get us too steep. So it has to be like that. And then we would get to a lower f of 10. Every time you have a slightly lower rate of change, then it kind of limits what happens to you. So remember, our slope can never be more than 7. So this part should be parallel. So this should be parallel to that right over there. This should be parallel. But we can never have a higher slope than that. So the way to max it out is to actually have a slope of 7. And so what is f of 10 going to be? So let's see, 10 minus negative 2, that is 12. Multiply both sides by 12, you get 84. So f of 10 minus 3 is going to be equal to 84. Or f of 10 is going to be equal to 87. So if you have a slope of 7, the whole way, you travel 12. That means you're going to increase by 84. If you started at 3, you increase by 84, you're going to get to 87.