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

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Example Questions

Example Question #11 : Calculus I — Derivatives

The speed of a car traveling on the highway is given by the following function of time:

f(t) = 5t^2+2t

Consider a second function:

g(t) = 10t+2

What can we conclude about this second function?

Possible Answers:

It represents the rate at which the speed of the car is changing.

It represents the change in distance over a given time .

It has no relation to the previous function.

It represents the total distance the car has traveled at time .

It represents another way to write the car's speed.

Correct answer:

It represents the rate at which the speed of the car is changing.

Explanation:

Notice that the function g(t) is simply the derivative of f(t) with respect to time. To see this, simply use the power rule on each of the two terms.

f'(t) = (2)(5)t + 2 = g(t)

Therefore, g(t) is the rate at which the car's speed changes, a quantity called acceleration.

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

Find the critical numbers of the function,

$f(x)=\frac{1+x^2}{x}$

Possible Answers:

$x =\left \{ -1,0,1\right \}$

$x =\left \{ -1\right \}$

$x =\left \{ 1\right \}$

$x =\left \{ -1,1\right \}$

Correct answer:

$x =\left \{ -1,1\right \}$

Explanation:

$f(x)=\frac{1+x^2}{x}$

1) Recall the definition of a critical point:

The critical points of a function f(x) are defined as points (a,f(a)) , such that x=a is in the domain of f(x) , and at which the derivative f'(a) is either zero or does not exist. The number x=a is called a critical number of f(x) .

2) Differentiate ,

$f'(x)=\frac{d}{dx}\left(\frac{1+x^2}{x} \right )=\frac{d}{dx}\left(\frac{1}{x}+x \right )=-\frac{1}{x^2}+1$

3) Set to zero and solve for ,

$f'(x)=-\frac{1}{x^2}+1= 0$

$\frac{1}{x^2}=1$

The critical numbers are,

$x =\left \{ -1,1\right \}$

We can also observe that the derivative does not exist at x = 0 , since the $-\frac{1}{x^2}$ term would be come infinite. However, is not a critical number because the original function f(x) is not defined at x=0 . The original function is infinite at x=0 , and therefore x = 0 is a vertical asymptote of f(x) as can be seen in its' graph,

Problem 7 plot

Further Discussion

In this problem we were asked to obtain the critical numbers. If were were asked to find the critical points, we would simply evaluate the function f(x) at the critical numbers to find the corresponding function values and then write them as a set of ordered pairs,

$\left \{(-1,-2), (1,2) \right \}$

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Example Question #2 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

The function f(x) is a continuous, twice-differentiable functuon defined for all real numbers.

If the following are true:

Which function could be f(x) ?

Possible Answers:

sin(x)

e^x

$\frac{1}{x^2}$

ln(x)

Correct answer:

ln(x)

Explanation:

To answer this problem we must first interpret our given conditions:

Implies the function is strictly increasing.
Implies the function is strictly concave down.

We note the only function given which fufills both of these conditions is lnx .

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Example Question #3 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

A jogger leaves City at t=0 . His subsequent position, in feet, is given by the function:

$x(t) = 5t^{2} - 3t + 2$ ,

where is the time in minutes.

Find the acceleration of the jogger at t=15 minutes.

Possible Answers:

$10\ ft/min$

$10\ ft/min^{2}$

$10\ ft$

$147\ ft/min^{2}$

$147\ ft/min$

Correct answer:

$10\ ft/min^{2}$

Explanation:

The accelaration is given by the second derivative of the position function:
a(t) = f''(t)

For the given position function:

$f(t) = 5t^{2} - 3t + 2$ ,

f'(t) = 10t - 3 ,

f''(t) = 10 .

Therefore, the acceleration at t=15 minutes is $10\ ft/min^{2}$ . Again, note the units must be in $ft/min^{2}$ .

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

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #11 : Calculus I — Derivatives

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

Example Question #2 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

Example Question #3 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

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