As we discussed earlier shows an object's position-versus-time graph. What is the velocity of the object at $t=6 \mathrm{~s}$?
A. $0.67 \mathrm{~m} / \mathrm{s}$
B. $0.83 \mathrm{~m} / \mathrm{s}$
C. $3.3 \mathrm{~m} / \mathrm{s}$
D. $4.2 \mathrm{~m} / \mathrm{s}$
E. $25 \mathrm{~m} / \mathrm{s}$

Assume that r and s are positive functions of t. If $r^2+s^4=25$ and $d r / d t=-3$, find $d s / d t$ when $s=2$.

A turntable with a radius of 8.0 cm rotates at 45 rpm. What is the linear speed of a point on the outside rim of the turntable? (A) 22.6 m/s (B) 5.6 m/s (C) 3.6 m/s (D) 0.38 m/s

A unity feedback system has a loop transfer function

$$L(s)=G_c(s) G(s)=\frac{K\left(s^2+0.1\right)}{s\left(s^2+1\right)}$$

Sketch the root locus as a function of $K$. Calculate where the segments of the locus enter and leave the real axis.

Using the following equations,

Ca(s) +2C(s) $\longrightarrow$ CaC$_2$(s) $\Delta$H = -628 kJ

CO$_2$(g) $\longrightarrow$ C(s) + O$_2$(g) $\Delta$H = 393.5 kJ

CaCO$_3$(s) + CO$_2$(g) $\longrightarrow$ CaC$_2$(s) + $\frac{5}{2}$O(g) $\Delta$H = 1538 kJ

determine the heat of reaction (in kJ) for

Ca(s) - C(s) + $\frac{3}{2}$O$_2$(g) $\longrightarrow$ CaCO$_3$(s)

Evaluate the following expressions if $p = 4, q = -2, r = 3$ and $s = -5$.

a) $2 p+4 q$

b) $5 r-3 s$

c) $3 q-4 s$

d) $6 p-8 q+4 s$

e) $3 r-3 p+5 q$

f) $p-q+r+s$

Is proposed as generating the set L of strings over {a, b} that contain equal numbers of a’s and b’s. If the grammar generates L, prove that it does so. If the grammar does not generate L, give a counterexample and prove that your counterexample is correct. In each grammar, S is the starting symbol. $S \rightarrow a S b|b S a| S S | \lambda$

Find the derivative the following ways: Using the Quotient Rule. Simplify your result. (b) By simplifying the quotient first. Verify that your answer agrees with part (a). $g(s)=\frac{6 s^{3}-15 s^{2}+9 s}{3 s}$

Let V be a vector space having dimension n, and let S be a subset of V that generates V. Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.) Let V be a vector space having dimension n, and let S be a subset of V that generates V. Prove that S contains at least n vectors.

Air at 100 kPa and $20^\circ C$ flows in a 12-cm-diameter pipe at a rate of 9.5 kg/min. The velocity of air in the pipe is (a) 1.4 m/s(b) 6.0 m/s(c) 9.5 m/s(d) 11.8 m/s(e) 14.0 m/s

An acorn falls from the top of an oak and accelerates at $9.8 \mathrm{~m} / \mathrm{s}^2$. It hits the ground in $1.5 \mathrm{~s}$. What is the speed of the acorn when it hits the ground?

A. $9.8 \mathrm{~m} / \mathrm{s}$

C. $15 \mathrm{~m} / \mathrm{s}$

B. $20 \mathrm{~m} / \mathrm{s}$

D. $30 \mathrm{~m} / \mathrm{s}$

A cube has side length s. If the side length increases by p units, what is the ratio of the surface area to volume of the new cube, in simplest form? A. $\frac{s}{s+p}$ B. $\frac{s+p}{6}$ C. $\frac{6}{p}$ D. $\frac{6}{s+p}$

A soccer ball is kicked at $37 ^ { \circ }$ above the horizontal, at 30.0 m/s by a professional player. Calculate and draw the trajectory of the ball from start till it hits the ground again. Draw vectors representing the velocity components and the acceleration at each of the following times: 0.0 s, 0.5 s, 2.0 s, 2.5 s, and 3.0 s.

What is the wheel's average angular velocity during the 10-s interval?
A. $0 \mathrm{rad} / \mathrm{s}$
B. $15 \mathrm{rad} / \mathrm{s}$
C. $30 \mathrm{rad} / \mathrm{s}$
D. $60 \mathrm{rad} / \mathrm{s}$