91Ó°ÊÓ

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).

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 Ball Thrown into the Air A ball is thrown straight up with an initial velocity of \(128 \mathrm{ft} / \mathrm{sec}\), so its height (in feet) after \(t\) sec is given by \(s=f(t)=128 t-16 t^{2}\). a. What is the average velocity of the ball over the time intervals \([2,3],[2,2.5]\), and \([2,2.1] ?\) b. What is the instantaneous velocity at time \(t=2\) ? c. What is the instantaneous velocity at time \(t=5 ?\) Is the ball rising or falling at this time? d. When will the ball hit the ground?

The symbol [ ] denotes the greatest integer function defined by \([x]=\) the greatest integer \(n\) such that \(n \leq x .\) For example, \([2.8]=2\), and \([-2.7]=-3 .\) In Exercises \(23-28\), use the graph of the function to find the indicated limit, if it exists. \(\lim _{x \rightarrow-1^{+}}[x]\)

Average Velocity of a Helicopter A helicopter lifts vertically from its pad and reaches a height of \(h(t)=0.2 t^{3}\) feet after \(t\) sec, where \(0 \leq t \leq 12\). a. How long does it take for the helicopter to reach an altitude of \(200 \mathrm{ft}\) ? b. What is the average velocity of the helicopter during the time it takes to attain this height? c. What is the velocity of the helicopter when it reaches this height?

The symbol [ ] denotes the greatest integer function defined by \([x]=\) the greatest integer \(n\) such that \(n \leq x .\) For example, \([2.8]=2\), and \([-2.7]=-3 .\) In Exercises \(23-28\), use the graph of the function to find the indicated limit, if it exists. \(\lim _{x \rightarrow-1}[x]\)

Prove the Constant Multiple Law for limits: If \(\lim _{x \rightarrow a} f(x)=L\) and \(c\) is a constant, then \(\lim _{x \rightarrow \alpha}[c f(x)]=c L .\)

a. Find the average rate of change of the area of a circle with respect to its radius \(r\) as \(r\) increases from \(r=1\) to \(r=2 .\) b. Find the rate of change of the area of a circle with respect to \(r\) when \(r=2\).

In Exercises 33-36, determine whether the function is continuous on the closed interval. \(f(x)=\sqrt{16-x^{2}}, \quad[-4,4]\)

Use the method of bisection to approximate the root of the equation \(x^{3}-x+1=0\) accurate to two decimal places. (Refer to Example 10.)

Let \(g\) be a continuous function on an interval \([a, b]\) and suppose \(a \leq g(x) \leq b\) whenever \(a \leq x \leq b .\) Show that the equation \(x=g(x)\) has at least one solution \(c\) in the interval \([a, b] .\) Give a geometric interpretation. Hint: Apply the Intermediate Value Theorem to the function \(f(x)=x-g(x)\)