1. Define the following objects:
a) Derivative f ¢(a) =
b) Definite integral òabf(x) dx
2.a) Use difference quotients (as in the definition of f ¢) to approximate f ¢(1) for f(x) = [Ö(8x+1)] accurate to 1 decimal place. Tell explicitly the numbers used in your calculation.
b) How do you know your estimate is accurate to one decimal place?
3. Let f(T) be the time, in minutes, that it takes for an oven to heat up to T 23Fahrenheit
a) What are the units of f ¢(T)?
b) What is the sign of f ¢(T)?
c) What is the meaning (in words) of f(300) = 10
d) What is the meaning (in words) of f ¢(300) = 0.1
4. A mudslide in California is pouring mud into a valley at a rate of h(t) ft3/hr where t is the number of hours since the mudslide began. Suppose that ò020h(t)dt = 2,000,000. Explain in practical terms what this equation means.
5. The table gives the windchill factor (23 F) as a function of the windspeed (in mph) when the air temperature is 2023 F.
| windspeed (mph) | 5 | 10 | 15 | 20 | 25 |
| windchill | 16 | 3 | -5 | -10 | -15 |
a) Give an approximation (including units) of the derivative of windchill with respect to windspeed when the air temperature is 2023 and the windspeed is 10 mph.
b) Explain in words what the derivative in (a) means.
6. Sketch the graph of g(x) -assuming that g(0) = 0. graph
7. You will see later in the semester that the derivative of [(x-3)/( x+2)] is the function [5/( (x+2)2)].
a)Use this fact to compute one of the following integrals (exactly):
ò14[5/( (x+2)2)]dx ò26[(x-3)/( x+2)]dx
b) Tell why you are able to compute the one but not the other at this time.
8. Give intervals for x in which f(x),
f ¢(x), and f ¢¢(x) have
values indicated. graph
f >0 , for x satisfying:
f ¢>0 , for x satisfying:
f ¢¢>0 , for x satisfying:
9. Sketch the graph of f and f ¢¢ given the graph of the derivative f ¢. graph
10. Consider the curve in the figure. graph
a) Give a left Riemann sum of three terms for ò1070f(t) dt
b) Sketch the estimate in (a) on the figure.
c) Give a right Riemann sum of three terms for ò1070f(t) dt
d) Based on your answers in (a) and (c), what is your best guess for ò1070f(t) dt
