AP Calculus AB : Corresponding characteristics of the graphs of ƒ, ƒ', and ƒ''

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #34 : Second Derivatives

On what intervals is the function  both concave up and decreasing?

Possible Answers:

Correct answer:

Explanation:

The question is asking when the derivative is negative and the second derivative is positive. First, taking the derivative, we get

Solving for the zero's, we see  hits zero at  and . Constructing an interval test,

, we want to know the sign's in each of these intervals. Thus, we pick a value in each of the intervals and plug it into the derivative to see if it's negative or positive. We've chosen -5, 0, and 1 to be our three values.

Thus, we can see that the derivative is only negative on the interval .

 

Repeating the process for the second derivative,

The reader can verify that this equation hits 0 at -4/3. Thus, the intervals to test for the second derivative are 

.  Plugging in -2 and 0, we can see that the first interval is negative and the second is positive.

Because we want the interval where the second derivative is positive and the first derivative is negative, we need to take the intersection or overlap of the two intervals we got:

 

If this step is confusing, try drawing it out on a number line -- the first interval is from -3 to 1/3, the second from -4/3 to infinity. They only overlap on the smaller interval of -4/3 to 1/3.

 

Thus, our final answer is 

Example Question #1 : Corresponding Characteristics Of The Graphs Of ƒ, ƒ', And ƒ''

On a closed interval, the function  is decreasing. What can we say about  and  on these intervals?

Possible Answers:

Two or more of the other answers are correct.

 is negative

 is decreasing

 is decreasing

 is negative

Correct answer:

 is negative

Explanation:

If  is decreasing, then its derivative is negative. The derivative of  is , so this is telling us that  is negative.

 

For  to be decreasing,  would have to be negative, which we don't know.

 

 being negative has nothing to do with its slope. 

 

For  to be decreasing, its derivative  would need to be negative, or, alternatively  would have to be concave down, which we don't know.

 

Thus, the only correct answer is that  is negative.

Example Question #1 : Corresponding Characteristics Of The Graphs Of ƒ, ƒ', And ƒ''

h(x)=\frac{f+g}{f}

If

and ,

then find .

Possible Answers:

Correct answer:

Explanation:

We see the answer is  when we use the product rule.

 

h(x)=\frac{f+g}{f}

h'(x)=\frac{(f'(x)+g'(x))(f(x))-(f(x)+g(x))(f'(x))}{f^{2}}

h'(x)=\frac{(2+4)(1)-(1+3)(2)}{1^{2}} =-2

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