224 AP calculus Exam slope at a point (x,y)

• MHB
• karush
In summary, the slope at a specific point (x,y) on the AP Calculus Exam is the rate of change of the function at that point, calculated by taking the derivative of the function at that point. To find the slope at a specific point (x,y), you need to take the derivative of the function and plug in the given values for x and y. The significance of the slope at a point (x,y) is that it represents the instantaneous rate of change of the function and can also be interpreted as the slope of the tangent line to the function at that point. The slope can be negative, indicating a decreasing function, and can provide information about the shape and behavior of the function on the overall graph.
karush
Gold Member
MHB
View attachment 9236
image to avoid typos

$f'(x)=\dfrac{2}{\left(x+2\right)^2}$
so then at slope $\dfrac{1}{2}$
$\dfrac{2}{\left(x+2\right)^2}=\dfrac{1}{2}$
isolate x
$4=(x+2)^2=x^2+4x+4$
then
$0=x(x+4)$
so
$x=0,-4$
thus the slope is $\dfrac{1}{2}$ at $(0,0), and \left(-4,2\right)\quad (C)$

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correct

karush said:
image to avoid typos

$f'(x)=\dfrac{2}{\left(x+2\right)^2}$
so then at slope $\dfrac{1}{2}$
$\dfrac{2}{\left(x+2\right)^2}=\dfrac{1}{2}$isolate x
$4=(x+2)^2=x^2+4x+4$
then
$0=x(x+4)$
so
$x=0,-4$
thus the slope is $\dfrac{1}{2}$ at $(0,0), and \left(-4,2\right)\quad (C)$
This is correct but at $4= (x+ 2)^2$ it is a little simpler to go
immediately
to $x+ 2= \pm 2$ so that $x+ 2= 2$, $x= 0$ and $x+ 2= -2$, $x= -4$.

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1. What is the slope at a point on the AP Calculus Exam?

The slope at a point on the AP Calculus Exam refers to the steepness of the curve at a specific point. It is calculated by finding the derivative of the function at that point.

2. How is the slope at a point calculated on the AP Calculus Exam?

The slope at a point is calculated by finding the derivative of the function at that point. This can be done using the limit definition of the derivative or through various differentiation rules.

3. What information does the slope at a point provide on the AP Calculus Exam?

The slope at a point provides information about the rate of change of the function at that point. It can also indicate the direction of the curve, whether it is increasing or decreasing.

4. Can the slope at a point be negative on the AP Calculus Exam?

Yes, the slope at a point can be negative on the AP Calculus Exam. This indicates that the function is decreasing at that point.

5. How is the slope at a point used in AP Calculus?

The slope at a point is an important concept in calculus and is used to solve various problems involving rates of change, optimization, and curve sketching. It is also used to find the equation of a tangent line at a specific point on a curve.

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