Math 2413 Lab 3 Name ______
(Dr. Khoury) Show all work Due as scheduled
1. Assume that f is continuous on its domain. Use the following information to draw the table of signs of , and then sketch the graph of f.
Domain: All real numbers except ;;
;
; / ;
;
Vertical asymptotes: ;
Horizontal asymptote: .
2. The graph of the derivative of a continuous function f is shown as follows:
a. On what intervals is f increasing?b. on what intervals is f decreasing? /
2
Fall 2015
c. At what point(s) does f has a local minimum?
d. At what point(s) does f has a local maximum?
e. On what intervals is f concaves downward?
f. On what intervals is f concaves upward?
g. State the point(s) of inflection.
h. Assume , , , , , , , , , . sketch the graph of f(x).
2
Fall 2015
3. The graph of the derivative of a continuous function f is shown as follows:
a. On what intervals is f increasing?b. On what intervals is f decreasing? /
2
Fall 2015
c. At what point(s) does f has a local minimum?
d. At what point(s) does f has a local maximum?
e. On what intervals is f concaves downward?
f. On what intervals is f concaves upward?
g. State the point(s) of inflection.
h. Assuming, sketch the graph of f(x).
2
Fall 2015
4. A particle moves according to a law of motion, , where t is measured in seconds and s in meters.
7
Fall 2015
a. Find the velocity function v at time t.
b. Find the critical numbers.
c. Find the acceleration function a at time t.
d. Find the points of inflection.
e. Complete the table of signs for s, v, and a, as we studied in class.
f. Graph s as we studied in class.
g. When in the particle at rest?
h. When in the particle is moving forward and when the particle is moving backward?
i. Find the total distance traveled during the first 8 seconds.
j. Draw a graph for all three functions s, v, and a.
k. When is the particle slowing down?
l. When is the particle speeding up?
7
Fall 2015
5. Given the ellipse and , .
7
Fall 2015
a. Find an equation of the tangent to the ellipse at the point where .
b. At what points is the tangent to the ellipse horizontal?
c. At what points is the tangent to the ellipse vertical?
d. Graph the ellipse and its tangent lines. Use your calculator with a window, , .
7
Fall 2015
6. A curve C is defined by the parametric equations, . Find the point at which C crosses itself for two different values of t. Then use your calculator with a window, , to graph C.
7. For the graph of f, find each of the following:
a. Draw the table of signs showing f, , and /7
Fall 2015
b. The critical number(s)
c. The point(s) of inflections
d. The interval(s) where f is decreasing
e. The interval(s) where f is increasing
f. The vertical asymptote(s), if any
g. The horizontal asymptote(s), if any
7
Fall 2015
8. For the given graphs, decide whether the Rolle’s Theorem can be applied or not. If it can, find the value(s) of c in the specified interval, state the reason(s) to why it cannot be applied.
Graph of f Graph of g
9. Use this graph to arrange the numbers in an increasing order./
10. An open box of maximum volume is to be made from a square piece of material, 24 inches on each side, by cutting squares from the corners and turning up the sides. Use calculus to find the dimensions of such box and its maximum volume. /
11. Which point(s) on the graph of is(are) closest to the point (0,2)? /
12. Let . Prove that for any interval , the value c guaranteed by the mean value theorem is the midpoint of the interval. Illustrate your answer by graphing a function with specific values a, b, and c.
13. Suppose that a missile is fired towards your location from 500 miles away and follows a flight path given by the parametric equations and , for . Two minutes later, you fire an interceptor missile from your location following the flight path and , for . Illustrate your answer by graphing the path of the two missiles.
a. Show that the interceptor missile will miss its target.
b. What can be done to the parametric equations so the interceptor missile will hit its target?
14. Find two numbers whose difference is 100 and whose product is a minimum.
15. Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.
16. A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
17. Use Newton’s method to approximate the root of the equation in the interval.
18. Complete the table of signs and draw the graph of each of the following functions exactly as we studied in class.
a. , .
b.
c.
d.
19. Use L’Hopital’s rule to evaluate the following limits:
a. b.
c. d.
7
Fall 2015