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Example Questions
Example Question #12 : Interpretations And Properties Of Definite Integrals
Example Question #13 : Interpretations And Properties Of Definite Integrals
Example Question #14 : Interpretations And Properties Of Definite Integrals
Example Question #15 : Interpretations And Properties Of Definite Integrals
Example Question #16 : Interpretations And Properties Of Definite Integrals
Example Question #17 : Interpretations And Properties Of Definite Integrals
Example Question #21 : Interpretations And Properties Of Definite Integrals
Example Question #21 : Interpretations And Properties Of Definite Integrals
Example Question #23 : Interpretations And Properties Of Definite Integrals
A pot of water begins at a temperature of and is heated at a rate of
degrees Celsius per minute. What will the temperature of the water be after 4 minutes?
Let denote the temperature of the pot after
minutes.
The first thing to realize is that the quantity is the derivative of
. Using the fundamental theorem of Calculus, we know that
From there, we just need to solve the integral. Letting u = t+1, du=dt, we have the following:
Where the second equality follows by the power rule, and re-substituting t+1 = u.
Thus, we now have the equation . Because we know that the water started at
, all we need to do is rearrange and substitute.
Yielding our final answer,
Example Question #1 : Second Derivatives
What is the concavity of at x=1?
Concave Down
Concave Horizontal
Concave Diagonal
Concave Up
No Concavity
Concave Up
First, find the double derivative of the function given.
You should get .
Next, plug in x=1 to get .
The double derivative is a positive number, which means the concavity is Concave Up.
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