Do the following for the function \( g(x)=10 x^{2}+9 x \). (a) Express the slope of the secant line in terms of \( x \) and \( h \). (b) Find \( m_{s e c} \) for \( h=0.5,0.1 \), and 0.01 at \( x=1 \). What value does \( m_{s e c} \) approach as \( h \) approaches 0 ? (c) Find the equation for the secant line at \( x=1 \) with \( h=0.01 \). (d) Graph \( g \) and the secant line found in part (c) on the same viewing window. (a) Type the slope of the secant line in terms of \( x \) and \( h \). \( m_{\text {sec }}=\square \) (Simplify your answer. Do not factor.)

It seems that I am unable to generate the graph directly. However, I can guide you on how to plot the functions using graphing software or a graphing calculator. ### Steps to Graph \( g(x) \) and the Secant Line: 1. **Graph the function \( g(x) = 10x^2 + 9x \)**: - This is a quadratic function that opens upwards. - You can plot points by substituting values of \( x \) into the function, such as \( x = -1, 0, 1, 2, 3 \). 2. **Graph the secant line \( y = 29.1x - 10.1 \)**: - This is a linear function with a slope of \( 29.1 \) and a y-intercept of \( -10.1 \). - You can plot points by substituting values of \( x \) into the equation, such as \( x = 0, 1, 2 \). 3. **Viewing Window**: - Set your x-axis to range from approximately \(-1\) to \(3\). - Set your y-axis to range from approximately \(-10\) to \(50\). ### Example Points to Plot: - For \( g(x) \): - \( g(-1) = 10(-1)^2 + 9(-1) = 10 - 9 = 1 \) - \( g(0) = 10(0)^2 + 9(0) = 0 \) - \( g(1) = 19 \) - \( g(2) = 10(2)^2 + 9(2) = 40 + 18 = 58 \) - For the secant line: - At \( x = 0 \): \( y = 29.1(0) - 10.1 = -10.1 \) - At \( x = 1 \): \( y = 29.1(1) - 10.1 = 19 \) - At \( x = 2 \): \( y = 29.1(2) - 10.1 = 38.1 \) By plotting these points and connecting them, you will see the curve of \( g(x) \) and the straight line of the secant line. If you have access to graphing software or a graphing calculator, you can input the functions directly to visualize them.