Exit Ticket for Day 29 of Calculus

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For the following problems, work this equation:

1. What is the derivative of the expression shown above?

Option 1

Option 2

Option 3

Option 4

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2. If I told you to find the "critical points" of that function, how would you do it?

You would need to determine if the second derivative of the function in INCREASING or DECREASING.

You would need to determine if the first derivative of the function in INCREASING or DECREASING.

You'd set the second derivative of the function equal to zero and then solve for the value(s) of x.

You'd set the first derivative of the function equal to zero and then solve for the value(s) of x.

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3. Find the 2 critical points of the shown shown above.

x = 1 and x = 6.33333

x = 1 and x = 7.5

x = 1 and x = 5.11111

x = 1 and x = -1

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4. If you knew that some value of x was a "critical point" of the function f(x), how could you determine whether that x was a RELATIVE MAXIMUM or a RELATIVE MINIMUM of the function f(x)?

If the second derivative of the function is POSITIVE at x, then this function has a minimum of f(x) at x.

If the second derivative of the function is NEGATIVE at x, then this function has a minimum of f(x) at x.

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5. If you look back at question 3, you'll notice that all of the options that I gave you have one of the two "critical points" of the function occurring at x=1. Therefore, you KNOW there is a critical point at that location. Is x=1 a RELATIVE MAXIMUM or a RELATIVE MINIMUM of this function?

relative maximum

relative minimum

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These are review questions!

6. Find the derivative of the following function:

Option 1

Option 2

Option 3

Option 4

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7. Suppose we know that f(1)=17 and f(10)=17. According to Rolle's theorem, we now also know _______.

the absolute maximum value of this function must be outside of the domain [1,10]

there is some value of x between 1 and 10 at which f'(x)=0 (notice the prime!)

there is some value of x between 1 and 10 at which f(x)=0 (notice there is no prime!)

the absolute maximum value of this function must be at either x=1 or x=10

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