Question

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use the figure to fill in the blanks in the following statements about the function f at point a.
(a) f( ) =
(b) f( ) =
(7,3) a
(7.2,3.8)
tangent line
f(x)
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Explanation:

Step1: Recall function - value definition

The value of the function $y = f(x)$ at a point $x=a$ is the $y$-coordinate of the point on the graph of the function with $x$-coordinate $a$. At point $A=(7,3)$ on the graph of $y = f(x)$, when $x = 7$, $y=f(7)=3$.

Step2: Recall derivative - slope relationship

The derivative of the function $y = f(x)$ at a point $x=a$, $f^{\prime}(a)$, is the slope of the tangent line to the graph of $y = f(x)$ at $x = a$. The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. The tangent - line to $y = f(x)$ at $x = 7$ passes through the points $(7,3)$ and $(7.2,3.8)$. So, $m=f^{\prime}(7)=\frac{3.8 - 3}{7.2 - 7}=\frac{0.8}{0.2}=4$.

Answer:

(a) $f(7)=3$
(b) $f^{\prime}(7)=4$