Math 161 - Test 4 Winter 2024 Instructions: This is a take-home test and is to be completed entirely on your own. Be sure to use calculus techniques to support your answers for each problem. Answers without the appropriate supporting work will not receive full credit. For any problem, you may use a calculator to complete algebraic operations (for example, to solve an equation), but not to complete calculus work (for example, to find where a function is increasing or decreasing). Again, all work on this test must be done on your own without any help from others; that includes tutors in the TLC or other locations, AI, and similar online help. (It is okay to get help from others on similar practice problems, but not on the problems found on the test.) You may consult information at the eLearning site, your notes and your textbook as necessary. If you are unsure of what I am asking you to do, you may contact me for clarification. You can contact me by stopping by my office, G-225, or by e-mail to pglutz@delta.edu. Please sign below to indicate that you have completed all work on your own following the instructions outlined above. Signature_ 1. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 7 in³/min. (12 points) Note: Vy =ër³h, Vcone = ±±ër³h cyl -лr²h 3 a) How fast the level in the pot rising when the coffee in the cone is 4 in deep? b) How fast is the level in the cone falling at that moment? 6" How fast is this level falling? How fast is this level rising? 2. (12 pts) Sketch a graph of a function satisfying the following conditions. Be sure to: • Label all points clearly with coordinates. • • f(0)=3 Show appropriate scale markings on both axes. Identify all asymptotes (with equation, if any). Identify all local extrema and inflection points. f'(2)=0 f'(0) does not exist f(2) = -1 f(3)=0 f'(x)>0 on (-∞,0) U(2,∞) lim f(x)=1 f'(x)<0 on (0,2) x--x f"(x)>0 on (-∞,0) U(0,3) U(5,∞) f"(x)<0 on (3,5) 3. a) Complete the first two rows of the table to verify that the function is one of the indeterminate forms (0°, ∞, or 1°). Complete the third row of the table to estimate the value of the limit 1 lim (secx+tan x) 6x. (4 pts) x→0+ 0.5 0.1 0.01 0.001 trend secx+tan x 1 6x 1 (secx + tan x) 6x 1 Use L'Hopital's rule to determine the exact value of the limit, lim (secx + tan x) 6x. (8 pts) b) Use L'Hopital's rule to evaluate the following limit. (8 pts) 5x2 lim x-0 cos(7x)-1 x→0+ 4. y=x(x+3)³, y' = 4x+9 4x+18 = y": 3(x+3) 9(x+3)³ a) For the function, find each of the following. (15 pts) x-intercept(s) y-intercept(s) critical values Intervals where the function is increasing Intervals where the function is decreasing Relative maximum(s) Relative minimum(s) Intervals where the function is concave up Intervals where the function is concave down Points of Inflection b) Below is a graph of the function. Use it to label the relative (local) minimum and maximum points as well as the inflection points. Be sure to give both x and y coordinates for each of these points. Round to 3 places after the decimal if necessary. (5 points) 10- -5 5. 5 c) If the function is restricted to the interval [-4, 5], what would be the absolute maximum and absolute minimum values? (4 points)