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In calculus, the derivative of a function at a certain point gives us crucial information. It tells the rate at which the function's value is changing at that point. Mathematically, the derivative of a function \( f(x) \) at a point \( x = a \) is denoted as \( f'(a) \).
For the given exercise, we have \( f'(3) = 2 \). This means the slope of the function at \( x = 3 \) is 2. In simpler terms, if you were to move slightly to either side of \( x = 3 \), the change in the function's value would be proportional to the distance moved, by a factor of 2.
Understanding derivatives is essential, not just for finding slopes, but also for knowing how functions grow, shrink, and change direction. A higher derivative value indicates a steeper slope in either the positive or negative direction.

A tangent line is a straight line that touches a function at a particular point and has the same slope as the function at that point. When you sketch a function, drawing the tangent line helps visualize how the function behaves near the point of tangency. In our exercise:

• The function value at \( x = 3 \) is \( f(3) = -2 \).
• The slope of the tangent line at \( x = 3 \) is given by the derivative value, \( f'(3) = 2 \).

Using the point-slope form of the line, we get the equation for the tangent line as:
\[ y = f(3) + f'(3) \cdot (x-3) \]
Substituting the given values:
\[ y = -2 + 2(x-3) \]
Simplifying, we find:
\[ y = 2x - 8 \]
This tangent line not only touches the function at \( x = 3 \), but also has the same slope as the function at that point. It provides a linear approximation of the function near \( x = 3 \).

Concavity tells us how a function curves around a point. Specifically, it tells us whether the function is curving upwards or downwards. In calculus, we determine this using the second derivative of the function. If the second derivative \( f''(x) \) is positive at a point, the function is concave up (shaped like a 'U'); if it's negative, the function is concave down (shaped like an upside-down 'U').

In our exercise, we have \( f''(3) = 3 \), which is greater than 0. This indicates that the function is concave up near \( x = 3 \). When sketching the function around \( x = 3 \), this means the curve will make an upward cup shape, further supporting the behavior determined by the tangent line.

• Concave up functions are often said to be 'holding water', as they curve upwards.
• Understanding concavity helps in predicting the shape and behavior of the function over intervals, not just at discrete points.

By looking at both the first and second derivatives, we get a comprehensive picture of the function's behavior at a given point, and this is particularly useful in sketching accurate graphs.