91Ó°ÊÓ

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=\frac{1}{x-2} ; \quad a=1\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=\left\\{\begin{array}{ll}x+2 & \text { if } x<3 \\ \ln (x-2)+5 & \text { if } x \geq 3\end{array}\right.\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 9} \sqrt{x}=3\)

The position function of an object moving along a straight line is given by \(s=f(t) .\) The average velocity of the object over the time interval \([a, b]\) is the average rate of change of f over \([a, b] ;\) its (instantaneous) velocity at \(t=a\) is the rate of change of \(\bar{f}\) at \(a .\) The position of a car at any time \(t\) is given by \(s=f(t)=\frac{1}{4} t^{2}\), \(0 \leq t \leq 10\), where \(s\) is given in feet and \(t\) in seconds. a. Find the average velocity of the car over the time intervals \([2,3],[2,2.5],[2,2.1],[2,2.01]\), and \([2,2.001] .\) b. Find the velocity of the car at \(t=2\).

Sketch the graph of the function \(f\) and evaluate (a) \(\lim _{x \rightarrow a^{-}} f(x)\), (b) \(\lim _{x \rightarrow a^{+}} f(x)\), and (c) \(\lim _{x \rightarrow a} f(x)\) for the given value of a. \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x<1 \\ 2 & \text { if } x=1 ; \quad a=1 \\ -x+2 & \text { if } x>1\end{array}\right.\)

Find the indicated limit. \(\lim _{x \rightarrow \pi} \sqrt{2+\cos x}\)

The position function of an object moving along a straight line is given by \(s=f(t) .\) The average velocity of the object over the time interval \([a, b]\) is the average rate of change of f over \([a, b] ;\) its (instantaneous) velocity at \(t=a\) is the rate of change of \(\bar{f}\) at \(a .\) Velocity of a Car Suppose the distance \(s\) (in feet) covered by a car moving along a straight road after \(t\) sec is given by the function \(s=f(t)=2 t^{2}+48 t\). a. Calculate the average velocity of the car over the time intervals \([20,21],[20,20.1]\), and \([20,20.01]\). b. Calculate the (instantaneous) velocity of the car when \(t=20 .\) c. Compare the results of part (a) with those of part (b).

Find the indicated limit. \(\lim _{x \rightarrow \pi / 4} \frac{\tan ^{2} x}{1+\cos x}\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=\left\\{\begin{array}{ll}\frac{x^{2}+x-6}{x-2} & \text { if } x \neq 2 \\\ 5 & \text { if } x=2\end{array}\right.\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 0}\left(x^{3}+1\right)=1\)