Exam 2: Key Concept: the Derivative

Every day the Office of Undergraduate Admissions receives inquiries from eager high school students.They keep a running account of the number of inquiries received each day, along with the total number received until that point.Below is a table of weekly figures from about the end of August to about the end of October of a recent year. Week di Inquiries That Week Total for Yeal 8/28-9/01 1085 11,928 9/04-9/08 1193 13,121 9/11-9/15 1312 14,433 9/18-9/22 1443 15,876 9/25-9/29 1588 17,464 10/02-10/06 1746 19,210 10/09-10/13 1921 21,131 10/16-10/20 2113 23,244 10/23-10/27 2325 25,569 One of these columns can be interpreted as a rate of change.Which one is it?

There is a function used by statisticians, called the error function, which is written y = erf (x).Suppose you have a statistical calculator, which has a button for this function.Playing with your calculator, you discover the following: x erf (x) 1 0.29793972 0.1 0.03976165 0.01 0.00398929 0.001 0.000398942 0 0 Using this information alone, give an estimate for erf'(0), the derivative of erf at x = 0 to 4 decimal places.

Estimate a formula for $f^{\prime}(x)$ for the function $f(x)=8^{x}$ .Round constants to 3 decimal places.

A concert promoter estimates that the cost of printing p full color posters for a major concert is given by a function Cost = c(p)where p is the number of posters produced. a)Interpret the meaning of the statement c(450)= 5400. b)Interpret the meaning of the statement c'(450)= 11.

Assume that f and g are differentiable functions defined on all of the real line.If f ' > 0 everywhere and f > 0 everywhere then must $\operatorname{lim}_{x \rightarrow+\infty} f(x)=\infty$ ?

A golf ball thrown directly upwards from the surface of the moon with an initial velocity of 20 meters per second and will attain a height of $s(t)=-0.8 t^{2}+20 t$ meters in t seconds.How fast is the golf ball going at its high point?

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

Given the following table of values for a Bessel function, $J_{0}(x)$ , estimate the derivative at x = 0.5. x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (x) 1.0 .9975 .9900 .9776 .9604 .9385 .9120 .8812 .8463 .8075

Assume that f and g are differentiable functions defined on all of the real line.Is it possible that f '(x)> g'(x)for all x and f(x)< g(x)for all x?

Draw the graph of a continuous function y = g(x)that satisfies the following three conditions: • g'(x)= 0 for x < 0 • g'(x)> 0 for 0 < x < 4 • g'(x)< 0 for x > 4

If $x(V)=V^{1 / 3}$ is the length of the side of a cube in terms of its volume, V, calculate the average rate of change of x with respect to V over the interval $3 < V < 4$ to 2 decimal places.

The derivative of $f(t)=e^{x}$ is $\pi e^{\pi-1}$ .