91Ó°ÊÓ

Find the slope of the tangent line to the graph of \(y=x^{2}\) at the point indicated and then write the corresponding equation of the tangent line. $$(-.4, .16)$$

A helicopter is rising straight up in the air. Its distance from the ground \(t\) seconds after takeoff is \(s(t)\) feet, where \(s(t)=t^{2}+t.\) (a) How long will it take for the helicopter to rise 20 feet? (b) Find the velocity and the acceleration of the helicopter when it is 20 feet above the ground.

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x & \text { for } x \neq 1 \\ 2 & \text { for } x=1 \end{array}\right.$$

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$$

If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x.\) $$f(x)=\frac{(6+x)^{2}-36}{x}, x \neq 0$$

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

A ball thrown straight up into the air has height \(s(t)=102 t-16 t^{2}\) feet after \(t\) seconds. (a) Display the graphs of \(s(t)\) and \(s^{\prime}(t)\) in the window \([0,7]\) by \([-100,200] .\) Use these graphs to answer the remaining questions (b) How high is the ball after 2 seconds? (c) When, during descent, is the height 110 feet? (d) What is the velocity after 6 seconds? (e) When is the velocity 70 feet per second? (f) How fast is the ball traveling when it hits the ground?

Refer to a line of slope \(m .\) If you begin at a point on the line and move \(h\) units in the \(x\)-direction, how many units must you move in the \(y\) -direction to return to the line? $$m=-3, h=.25$$

Let \(C(x)\) be the cost (in dollars) of manufacturing \(x\) bicycles per day in a certain factory. Interpret \(C(50)=5000\) and \(C^{\prime}(50)=45\).

We specify a line by giving the slope and one point on the line. We give the first coordinate of some points on the line. Without deriving the equation of the line, find the second coordinate of each point. Slope is \(-3, (2,2)\) on line; \((3,) ;(4,) ;(1,)\)