Question 21 A particle starts from the origin with a velocity \( 5 \mathrm{~cm} / \mathrm{sec} \) and moves in a straight line, its acceleration at time \( t \) seconds being \( 3 t^{2}-5 t \mathrm{~cm} / \mathrm{sec}^{2} \), what will be the velocity of the particle? \( \begin{array}{ll}\text { (a) } 28 \mathrm{~cm} / \mathrm{sec} & \text { (b) } 29 \mathrm{~cm} / \mathrm{sec} \\ \text { (c) } 30 \mathrm{~cm} / \mathrm{sec} & \text { (d) } 31 \mathrm{~cm} / \mathrm{sec}\end{array} \)

Show that \( F(x)=|x| \) is continuous everywhere.

8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \).

Suppose \( f \) is continuous on \( [1,5] \) and the only solutions of the equation \( f(x)= \) 6 are \( x=1 \) and \( x=4 \). If \( f(2)=8 \), explain why \( f(3)>6 \). Show that \( F(x)=|x| \) is continuous evervwhere.

7. Find \( \lim _{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \) 8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \)

11. Find the values of \( a \) and \( b \) that make \( f \) continuous everywhere. \[ f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 & \text { if } 2 \leq x<3 \\ 2 x-a+b & \text { if } x \geq 3\end{array}\right. \]