Test 1 review questions
1. If : Evaluate f(2), f(a), f(2+a), and .
2. The population of Rockville was 47,000 in 2000, and grew to 61,000 in 2010.
a. Assuming the population grew linearly, find a formula for p, the population of the city as a function of t, the number of years since 2000.
b. Interpret the slope of your answer to part (a) in practical terms (ie your answer should include units and a description of what it tells you in the context of the problem)
The slope of 1400 represents the average increase in population each year over the given decade.
c. Now assume that the population of Rockville grew exponentially. Find a formula for p, the population of the city as a function of t, the number of years since 2000.
3. If , find each of the following:
4. Express as a composition of functions
5. For the function f(x) graphed to the right:
What is the domain? The range?
Domain, [-3,3],Range [-4 5]
6. Zoom in on the graph of near the point (2, 8) until it looks like a straight line.
Then use the trace feature to find the coordinates of another point on the curve, and find the slope of the line connecting those 2 points. Finally, find an equation for that line.
7. Let P be the point (2, 8). Find the slope of the secant line PQ, where Q is the point for the following values of x (Give answers to 3 decimal places)
a) 3 b) 2.1 c) 2.001
a) 19; b) 12.61; c) 12.006
d) Using parts (a)-(c) above, and additional points if necessary, approximate the slope of the tangent line to at the point x=2.
The limit of the above slopes appears to be 12, so this should be the slope of the tangent line.
e) What is the connection between this problem and #9?
This is a numerical approximation to the slope of the same line that we found graphically in 9.
8. Use the definition of the derivative to find the derivative of at (2,-2). Then use this to find the tangent line to this function at that point
So the tangent line goes through (2,-2) with slope -3, which is y=-3x+4
9. f(x) is the function graphed to the right:
Yes, it is 2
b. Does f(-1) exist? If so, what is it?
Yes, it is 4.
c. Is f continuous at x= -1? Why or why not?
No, the limit and the value at that point are not equal
Yes, it is 0
e. Does f(1) exist? If so, what is it?
Yes, it is 0
f. Is f continuous at x= 1? Why or why not?
Yes, the answers to parts (d) and (e) are equal.
10. Evaluate the following limits. Use numeric and graphic means, and where possible also use algebra to confirm the answer:
Only the algebraic solutions are shown . You should
approximate these numerically and graphically too
This does not exist. If x approaches 2 (from either above or below) the function grows to ∞.
11. Find a number b such that is continuous everywhere.
For this to be continuous at x=2, the limit from the left, limit from the right, and the value must all agree. The value at x=2 is 22=4, so we need the limit to be 4. Since x2is a continuous function, the limit from the left (the domain for that piece) is going to be 4, but we need to make sure that the limit from the right is also 4. As x approach 2, x+b approaches 2+b, so we need 2+b=4, or b=2.
12. The height of a ball, t seconds after it is thrown, is given by .
a. What is the average speed of the ball over the first second after it is thrown?
b. What is the instantaneous speed of the ball one second after it is thrown?
13. The graph shows f, f, and f. Which curve is which? How do you know?
The blue is f, its derivative f is in red, and reds derivative is f in green.
When the blue curve is increasing the red curve is positive, and when it is decreasing the red is negative. Also, when the blue is concave down, the red (its derivative) is decreasing and the green (its 2nd derivative) is negative, while when the blue is concave up the red is increasing and the green is positive.