Exam 2: Key Concept: the Derivative

Given the following data about a function f, estimate $f^{\prime}(4.75)$ . x 3 3.5 4 4.5 5 5.5 6 f(x) 10 8 7 4 2 0 -1

Is the graph of $f(x)=\frac{1}{x+9}$ continuous at x = -9?

To study traffic flow along a major road, the city installs a device at the edge of the road at 1:00a.m.The device counts the cars driving past, and records the total periodically.The resulting data is plotted on a graph, with time (in hours since installation)on the horizontal axis and the number of cars on the vertical axis.The graph is shown below; it is the graph of the function C(t)= Total number of cars that have passed by after t hours.When is the traffic flow greatest?

The length, x, of the side of a cube with volume V is given by $x(V)=V^{1 / 3}$ .Is the average rate of change of x with respect to V increasing or decreasing as the volume V decreases?

A horticulturist conducted an experiment to determine the effects of different amounts of fertilizer on the yield of a plot of green onions.He modeled his results with the function $Y(x)=-0.5(x-2)^{2}+5$ where Y is the yield in bushels and x is the amount of fertilizer in pounds.What are $Y(0.75)$ and $Y^{\prime}(0.75)$ ? Give your answers to two decimal places, specify units.

One of the following graphs is of $f(x)$ , and the other is of $f^{\prime}(x)$ .Is $f(x)$ the first graph or the second graph?

Let t(h)be the temperature in degrees Celsius at a height of h meters above the surface of the earth.What does t(h)+ 15 mean in practical terms?

Suppose that f(T)is the cost to heat my house, in dollars per day, when the outside temperature is T ${ } ^ { \circ } F$ .If f(28)= 11.10 and f '(28)= -0.12, approximately what is the cost to heat my house when the outside temperature is 25 ${ } ^ { \circ } F$ ?

Find the derivative of $m(x)=3 x^{3}$ at x = 1 algebraically.

The definition of the derivative function is $f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}$

Estimate $\lim _{h \rightarrow 0} \frac{(6+h)^{2}-36}{h}$ to 2 decimal places by substituting smaller and smaller values of h.

Could the Function 1 be the derivative of the Function 2? Function 1 Function 2

Could the first graph, A be the derivative of the second graph, B? A B

For $f(x)=2^{-x}$ , estimate $f^{\prime}(0)$ to 3 decimal places.

Let f(x)= log(log(x)).Estimate $f^{\prime}(7)$ to 3 decimal places using any method.

The height of an object in feet above the ground is given in the following table.Compute the average velocity over the interval 1 $\le$ t $\le$ 3. t (sec) 0 1 2 3 4 5 6 y (feet) 10 45 70 85 90 85 70

Alone in your dim, unheated room you light one candle rather than curse the darkness.Disgusted by the mess, you walk directly away from the candle.The temperature (in ${ } ^ { \circ } F$ )and illumination (in % of one candle power)decrease as your distance (in feet)from the candle increases.The table below shows this information. distance(feet) Temp. illumination 0 56 100 1 55.5 85 2 54.5 75 3 53 67 4 51 60 5 48 56 6 44.5 53 What is the average rate at which the temperature is changing (in degrees per foot)when the illumination drops from 75% to 56%? Round to 2 decimal places.

Assume that f and g are differentiable functions defined on all of the real line.If f'(x)= g'(x)for all x and if

Find the derivative of $g(x)=2 x^{2}+8 x-6$ at x = 4 algebraically.

The height of an object in feet above the ground is given in the following table.If heights of the object are cut in half, how does the average velocity change, over a given interval? t (sec) 0 1 2 3 4 5 6 y (feet) 10 45 70 85 90 85 70