Exam 2: Limits and Derivatives

If a ball is thrown into the air with a velocity of 80 ft/s, its height in feet after t seconds is given by s(t) = 80t - 16t². It will be at maximum height when its instantaneous velocity is zero. Find its average velocity from the time it is thrown (t = 0) to the time it reaches its maximum height.

Given the graph of $y = f ^ { \prime } ( x )$ below, select a graph which could be that of y = f (x).

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car's velocity is given in the table below: t(s) 0 1 2 3 4 5 6 7 8 v(t)(ft/s) 0 18 47 77 104 132 163 184 208 (a) Find the car's average acceleration for the following time intervals: (i) [4; 6] (ii) [4; 5] (iii) [3; 4] (b) Estimate the car's acceleration at t = 4.

Consider the function $f ( x ) = \frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 }$ Make an appropriate table of values in order to determine the indicated limits: (a) $\lim _ { x \rightarrow - 3 ^ { + } } f ( x )$ (b) $\lim _ { x \rightarrow - 3 ^ { - } } f ( x )$ (c) Does $\varliminf _ { x \rightarrow - 3 } f ( x )$ exist? If so, what is its value? If not, explain.

A weight is attached to a spring. Suppose the position (in meters) of the weight above the floor t seconds after it is released is given by $P ( t ) = 0.5 \sin \left( \pi t + \frac { n } { 2 } \right) + 1.2$ (a) What is the position of the weight when t = 2? When t = 3? When t = 4? (b) What is the average rate of change of the position of the weight (in m/s) over the time interval [2, 4]? Over the time interval [2, 3]? (c) The average rate of change of the position of the weight over the time period [2; 6] is 0. Does this mean that the weight has come to a stop? Why or why not?

Find the constant(s) c that make(s) the function $f ( x ) = \left\{ \begin{array} { l l } c ^ { 2 } - x ^ { 2 } & \text { if } x < 2 \\2 ( c - x ) & \text { if } x \geq 2\end{array} \right.$ continuous on $( - \infty , \infty )$ .

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$

The point $P ( 1 , \sqrt { 3 ) }$ lies on the curve $y = \sqrt { 4 - x ^ { 2 } }$ Let Q be the point $\left( x , \sqrt { 4 - x ^ { 2 } } \right)$ (a) What is the slope of the secant line PQ (correct to 6 decimal places) for the following values of x? (i) 2 (ii) 1.5 (iii) 1.1 (iv) 1.01 (v) 1.001 (vi) 0 (vii) 0.5 (viii) 0.9 (ix) 0.99 (x) 0.999 (b) Use your results from part (a) to estimate the slope of the tangent line to the graph of $y = \sqrt { 4 - x ^ { 2 } \text { at } x = 1 }$

Find the value of the limit $\lim _ { x \rightarrow - \infty } \left( \sqrt { x ^ { 2 } + 2 x } - 2 x \right)$

At what value(s) of x is the function $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + 4 x + 5 & \text { if } x < - 2 \\\frac { 1 } { 2 } x & \text { if } - 2 \leq x \leq 2 \\1 + \sqrt { x - 2 } & \text { if } x > 2\end{array} \right.$

-For the function whose graph is given above, determine $\lim _ { x \rightarrow 2 } f ( x )$

Find the limit.(a) $\lim _ { x \rightarrow \infty } \frac { 1 } { x ^ { 2 } }$ (b) $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - \sqrt { x ^ { 2 } - x } \right)$ (c) $\lim _ { x \rightarrow - \infty } \left( x - 3 x ^ { 2 } \right)$

Find the value of the limit $\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 }$

Find the value of the limit $\lim _ { x \rightarrow 1 } \left( x ^ { 17 } - x + 3 \right)$

Given the following information about limits, select a graph which could be the graph of y = f (x). f(x)=f(x)=0f(x)=f(x)=\infty f(x)=f(x)=-\infty

Given the following information about limits, select a graph which could be the graph of y = f (x). f(x)=f(x)=0f(x)=f(x)=\infty f(x)=f(x)=-\infty

Using the graph below, determine the following: (a) $\lim _ { x \rightarrow \infty } f ( x )$ (b) $\lim _ { x \rightarrow 2 ^ { + } } f ( x )$ (c) $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$ (d) $f _ { x \rightarrow - \infty } f ( x )$ (e) $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ (f) $\lim _ { x \rightarrow 0 ^ { - } } f ( x )$ (g) Find the horizontal asymptote(s) of the graph of y = f (x).(h) Find the vertical asymptote(s) of the graph of y = f (x).

-For the function whose graph is given above, determine $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$

The definition of continuity of f (x) at a point requires three things. List these three conditions, and in each case give an example (a graph or a formula) which illustrates how this condition can fail at x = a.

At what value(s) of x does the function $\frac { ( x + 1 ) ^ { 2 } } { x ^ { 2 } - 1 }$ have a removable discontinuity?