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**wren17** 1. The limit, as x tends to 2, of the function $\displaystyle y=\frac{x^2 + bx - 28}{x - 2}$ exists, and is equal to 16. What is the value of the whole number coefficient b?

2. The first derivative of $\displaystyle y = (6x^3 + 11)^9$ is of the form $\displaystyle ax^2(6x^3 + 11)^n$ , where a is a positive number. What is the value of the whole number coefficient a?

3. For the ellipse $\displaystyle 4x^2 + 9y^2 = 724$, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

4. The number I is defined as $\displaystyle \int_{1}^{2} 4~sin~x~cos~xdx$. The value of I, correct to the nearest hundredth, is ______________.

5. A spring has a motion that can be modelled by the differential equation $\displaystyle \frac{d^2 y}{dt^2} = -5y$. One solution of this equation is $\displaystyle y = 4~cos~kt$, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:

$\displaystyle v(t) = 20sin[\frac{\pi t}{60}]~~~~~~~~0 \geq t \geq 30$

$\displaystyle v(t) = 20 ~~~~~~~30 \geq t \geq 100$

$\displaystyle v(t) = -0.05(t - 100)^2 + 20~~~~~~~~100 \geq t \geq 120$

a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

b. Determine the total distance travelled by the train in the 120-second time interval.

4. Let the function f be defined as $\displaystyle f(x) = \frac{tan~x - sin~x}{x^3}$.

a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.

b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.