91Ó°ÊÓ

Evaluating a Limit by Direct Substitution Exercises \(45-64\) , find the limit by direct substitution. $$\lim _{x \rightarrow 1} \arccos \frac{x}{2}$$

Determining Convergence or Divergence In Exercises \(65 - 68\) , create a scatter plot of the terms of the sequence. Determine whether the sequence converges or diverges. When it converges, estimate its limit. $$ a _ { n } = \frac { 3 \left[ 1 - ( 1.5 ) ^ { n } \right] } { 1 - 1.5 } $$

The profit \(P\) (in hundreds of dollars) that a company makes depends on the amount \(x\) (in hundreds of dollars) the company spends on advertising. The profit function is \(P(x)=200+30 x-0.5 x^{2}.\) Using your knowledge of the slopes of tangent lines, show that the profit is increasing on the interval \([0,20]\) and decreasing on the interval \([40,60] .\)

Consider the function \(f(x)=3 x^{2}-2 x.\) (a) Use a graphing utility to graph the function. (b) Use the trace feature to approximate the coordinates of the vertex of this parabola. (c) Use the derivative of \(f(x)=3 x^{2}-2 x\) to find the slope of the tangent line at the vertex. (d) Make a conjecture about the slope of the tangent line at the vertex of an arbitrary parabola.