MCS 118 Quiz 1. Fall (5pts) Solve the following equations for x. 7x 2 = 4x x 2 5x = 2

Transcription

1 MCS 8 Quiz Fall 6. (5pts) Solve the following equations for. 7 = (5pts) Solve the following equations for. 3 5 = 3. (5pts) Factor as much as possible.

2 4. (5pts) Simplify (5pts) Solve the following inequality giving your solution in interval notation. 5 > (5pts) Solve the following inequality giving your solution in interval notation and sketching it on a number line. 3 +

3 7. (5pts) Suppose that f() = (a) Compute f(4). (b) Compute f(). (c) Compute f( + h). 8. (5pts) Find the equation of the line containing the points (, 4) and (4, 4).

4 9. (5pts) What is the domain of the function +?. (5pts) Solve the following inequality giving your solution in interval notation. ( 3)( + ) <

5 MCS 8 Quiz Fall 6. (5 pts) Match each of the functions below to class that best describes it by drawing a line from the function to the corresponding class. function f () = f () = π f 3 () = ( )( + ) f 4 () = f 5 () = function class constant linear quadratic cubic rational f 6 () = +4 polynomial. (6 pts) Let f() =, g() = /, and h() = 4 3. Find a formula for each of the following functions. Please simplify your answers when appropriate. (a) (f h)() (b) (fg)() (c) ( ) h () g

6 3. (6 pts) (a) Let f() = and g() = /. Compute (f g)(). Do not simplify. (b) Complete the following table. f() g() (g f)() (6 pts) Each of the functions given below can be written as the composition f() = g(h()). Identify g() and h(). (a) f() = ( 3) 4 (b) f() = ( ) + ( ) + 3

7 5. (8 pts) Use the graph of the function f() illustrated below to graph the following. Describe the relationship between the new graph and the original graph. (a) y = f( ) + 3 (b) y = f( )

8 6. (4 pts) Suppose f() = 4 8. Find f (). 7. (6 pts) Are the functions illustrated below invertible? Why or why not.

9 MCS 8 Quiz 3 Fall 6. ( pts) Use the graph of the function f() illustrated below to evaluate the following limits (a) lim f() (b) lim f() + (c) lim f() (d) lim f(). (7 pts) Let f() =. (a) Use your calculator to complete the following table f()

10 (b) Based on your calculations above, what do you think is the value of lim f()

11 3. (5 pts) Evaluate each of the following limits. (a) lim 3 + (b) lim (c) lim 3 5 ( 3)

12 4. (6 pts) Draw a graph of a function f() such that all of the following properties hold. lim f() = 5 3 lim f() = 3+ f(3) = 5 5. ( pts) Let f() = 5( ) 3, c = 3, L = 4, and ǫ =.. (a) Find lim c f(). (b) Find a δ > such that the graph of y = f() leaves the window a δ < < a + δ, b ǫ < y < b + ǫ, by the side and not through the top or bottom. (c) Sketch the graph here. Label the boundaries and the centers of the intervals for and y from your graphing calculator s window. y

13 MCS 8 Quiz 4 Fall 6. (7 pts) Use the definition of the derivative to compute f () for f() =.. (6 pts) Compute the derivative of each of the following using the derivative rules. (a) y = (b) y = 4 (c) g() = (d) f() = 7

14 3. (8 pts) Match graphs of derivatives in the right hand column to function s graph in the left hand column. Function Derivative

15 4. (6 pts) Find the equation of the tangent line to the graph of y = at the point (, 3). 5. (6 pts) For the function f() illustrated in the figure below, indicate the value(s) of where f() is either not continuous or not differentiable f() is not continous at. f() is not differentiable at.

16 6. (7 pts) Barbara is walking up and down Seventh St. on her way home from school. Tom is standing at the corner of Seventh St.and College Ave. graphing her velocity with a radar gun. Positive velocities indicate northward motion. Below is a graph of Barbara s velocity as a function of time. Use the graph to fill in the blanks or choose the correct word in the following story. In between t = and t = Barbara s velocity is { increasing, decreasing } and she is moving { away from, towards } Tom. In between t = and t = she is moving at a constant velocity. Then she starts to { slow down, speed up }. At time t = she realizes she has forgotten something so she turns around and starts heading { away from, towards } Tom.

17 MCS 8 Quiz 5 Fall 6. (6 pts) Use the definition of the derivative to compute f () for f() = 5.. (6 pts) Let f() = f () = f () = 6 6 Find the critical points of f() and determine whether each is a local ma or local min.

18 3. ( pts) Compute the derivative of each of the following using the derivative rules. (a) y = /5 6 (b) y = 4 3 (c) g() =

19 4. (8 pts) Below is the graph of f() = (a) Draw the secant line between the points (, f()) and (5, f(5)). (b) Compute the slope m of the secant line between the points (, f()) and (5, f(5)). (c) Draw a tangent line to the graph of y = f() having the same slope as the secant line. (d) Estimate the value of such that f () = m where m is the slope you computed in part (b).

20 5. (8 pts) Use the graph provided to complete each of the statements below by circling the word that best completes the sentence. (a) At the point marked A, f() is increasing, decreasing, neither. (b) At the point marked A, f () is positive, negative, zero. (c) At the point marked A, f() is concave up, concave down, neither. (d) At the point marked B, f() has a local maimum, local minimum, neither. (e) At the point marked B, f () is positive, negative, zero. (f) At the point marked B, f () is positive, negative, zero. (g) At the point marked C, f() has a local maimum, local minimum, neither. (h) At the point marked C, f () is positive, negative, zero. (i) At the point marked C, f () is positive, negative, zero.

21 MCS 8 Quiz 6 Fall 6. (6 pts) Below are the graphs of a function f(), its derivative f (), and its second derivative f (). Identify them y y y 3

22 . ( pts) Compute the derivative of each of the following using the derivative rules. Do not simplify. (a) y = (b) y = (c) y =

23 3. (6 pts) Evaluate the following limits. (a) lim (b) lim (6 pts) Find a possible formula for each of the graphs shown below.

24 5. ( pts) Suppose you have to graph y = ( )( + ) ( + 3) ( 3) = (a) Is there a horizontal asymptote? If so, what is it? If not, why not? (b) What is the y-intercept? (c) What are the vertical asymptotes? (d) What are the -intercepts?

25 6. (8 pts) A farmer has 4 feet of fencing. He is making a rectangular pig pen with 5 different rectangular components (see figure below). Use calculus to determine the dimensions of the pen of maimum area.

26 MCS 8 Midterm Eam Fall 6. (8 pts) Factor the following polynomials if possible. (a) f() = 5 (b) f() = (6 pts) Simplify h +h h. 3. (4 pts) Suppose that f() = and g() = 3. Find f(g()). (do not simplify.)

27 4. (9 pts) If h() = ( 3) 4, find two functions f and g such that h() = f(g()). f() = g() = 5. (8 pts) Suppose f() = 3 +. What is a formula for f ()? 6. ( pts) Compute each of the following limits. ( + )( 4) (a) lim + + (b) lim 3 ( 3)( 5) ( 3)

28 7. (8 pts) Let g() = ( + 4)( ). ( 5) (a) What are the roots of g()? (b) At what values of is g discontinuous? (c) On the number line below, indicate all intervals on which g() is positive, and the intervals on which g() is negative. 8. (6 pts) Use the graph of the function f() illustrated below to evaluate the following limits. If the limit does not eist then eplain why y lim f() = lim f() = + lim f() = lim f() =

29 9. ( pts) Here is the graph of a function, repeated several times. Sketch the given lines and find their slopes. (a) The secant line connecting the point (, f()) to ( +., f( +.)) (b) The secant line connecting the point (, f()) to ( + (.), f( + (.))) (c) The tangent line at the point (, f()) (a) (b) (c) ( pts) Let f() = 5 +. (a) f( ) = (b) f( + h) = (c) f( + h) f( ) = (please simplify) (d) Use your results above to compute f ( ).

30 . (9 pts) Let f() =, c =, L = /, and ǫ =.. + (a) Find lim c f(). (b) Find a δ > such that the graph of y = f() leaves the window a δ < < a + δ, b ǫ < y < b + ǫ, by the side and not through the top or bottom. (c) Sketch the graph here. Label the boundaries and the centers of the intervals for and y from your graphing calculator s window. y