Question

find the indicated quantities for f(x) = 3x². (a) the slope of the secant line through the points (1, f(1)) and (1 + h, f(1 + h)), h ≠ 0 (b) the slope of the graph at (1, f(1)) (c) the equation of the tangent line at (1, f(1)) (a) the slope of the secant line through the points (1, f(1)) and (1 + h, f(1 + h)), h ≠ 0, is 6 + 3h. (b) the slope of the graph at (1, f(1)) is \boxed{}. (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the definition of the slope of the graph (derivative)

The slope of the graph of a function \( f(x) \) at a point \( x = a \) is the limit of the slope of the secant line as \( h \to 0 \). From part (A), we have the slope of the secant line through \( (1, f(1)) \) and \( (1 + h, f(1 + h)) \) is \( 6 + 3h \). To find the slope of the graph at \( x = 1 \), we take the limit as \( h \to 0 \) of \( 6 + 3h \).

Step2: Evaluate the limit

We calculate \( \lim_{h \to 0} (6 + 3h) \). As \( h \) approaches 0, the term \( 3h \) approaches 0. So, \( \lim_{h \to 0} (6 + 3h)=6 + 3(0)=6 \).

Answer:

6