Explain the geometric meaning of the difference quotient faund in this exercise? Let \( g(x)=x^{2}=-x^{3}+x \). Find and simplify the difference quotient \( g(x)-g(a) \) \( x-a \).

We begin by recalling that the difference quotient \[ \frac{g(x)-g(a)}{x-a} \] represents the slope of the secant line joining the points \((a, g(a))\) and \((x, g(x))\) on the graph of \(g\). As \(x\) approaches \(a\), this quotient approaches the derivative \(g'(a)\), which is the slope of the tangent line at the point \((a, g(a))\). Assume first that the exercise considers the function \[ g(x)=x^{2}. \] Then the difference quotient is \[ \frac{g(x)-g(a)}{x-a}=\frac{x^{2}-a^{2}}{x-a}. \] Notice that \(x^{2}-a^{2}\) is a difference of two squares and factors as \[ x^{2}-a^{2}=(x-a)(x+a). \] Dividing by the common factor \((x-a)\) (with the restriction that \(x\neq a\)) we obtain \[ \frac{(x-a)(x+a)}{x-a}=x+a. \] Thus the simplified difference quotient is \[ x+a. \] Geometrically, this means that for any \(x\) (with \(x\neq a\)), the slope of the secant line connecting \((a,a^2)\) and \((x,x^2)\) is \(x+a\). As \(x\) gets closer to \(a\), the secant line approaches the tangent line at \(a\), and its slope tends to \(2a\) (since \(\lim_{x\to a}(x+a)=2a\)), which is the derivative of \(x^2\) at \(a\). Alternatively, if the intended function is \[ g(x)=-x^{3}+x, \] we can compute the difference quotient for that case as well. The difference quotient becomes \[ \frac{g(x)-g(a)}{x-a} = \frac{\bigl[-x^{3}+x\bigr] -\bigl[-a^{3}+a\bigr]}{x-a} = \frac{-x^{3}+x+a^{3}-a}{x-a}. \] We can group the terms to write \[ \frac{-\left(x^{3}-a^{3}\right) + (x-a)}{x-a}. \] Recall that the difference of cubes factors as \[ x^3-a^3=(x-a)(x^2+ax+a^2). \] Thus, \[ \frac{-\left(x-a\right)(x^2+ax+a^2) + (x-a)}{x-a} =\frac{(x-a)\left[-(x^2+ax+a^2)+1\right]}{x-a}. \] Canceling the common factor \((x-a)\) (again, with \(x\neq a\)), we obtain \[ -(x^2+ax+a^2)+1, \] or equivalently \[ 1 - (x^2+ax+a^2). \] In this situation, the difference quotient \(1 - (x^2+ax+a^2)\) represents the slope of the secant line connecting the points \((a, g(a))\) and \((x, g(x))\) on the curve \(y=-x^{3}+x\). As \(x\) approaches \(a\), this expression will approach the derivative \(g'(a)\), which is the slope of the tangent line at \(a\). In summary, the geometric meaning of the difference quotient is that it gives the slope of the secant line between two points on the graph of the function \(g\). Whether you choose \(g(x)=x^{2}\) or \(g(x)=-x^{3}+x\), its limit as \(x\) approaches \(a\) yields the slope of the tangent line at \(x=a\).