Limits and Continuity
When does a limit not exist?
1. A limit does not exist if it approaches positive or negative infinity or if it does not converge on a given value.
What is the derivative of a linear function?
2. The derivative of a linear function is its slope
When f(x) has a point of inflection, what does f'(x) and f"(x) have?
3. f'(x) has a relative max or min and f"(x) equals zero and changes sign
int 5x3 + 2
4.
(5x^4/4) +2x
Rotate the solid whose base is bounded by the x-axis and f(x) around the x-axis from the bounds x=0 and x=3.
f(x) = 1/2 x
5.
2.25 pi or 7.069
What is Ms. Braney's favorite animal?
6. Elephants
What is the one removable discontinuity and what are the two non-removable discontinuities?
7. Removable: holes
Non-removable: jumps and asymptotes
Derivative of 12x2 - cos(x)
8. 24x + sin(x)
Determine the critical values of the function
f(x)=3x^3-6x^2
9.
x = 0 or 4/3
int 15x^2(5x^3+ 1)^2 dx
10.
1/3 (5x^3 + 1)^3 + c
Find the area of the region bounded by the graphs of f(x) and g(x) from x=-1 and x=1
f(x) = -X^2 + 2X + 4
g(x) = 2X + 2
11. 3.333
What state did Ms. Braney go to college in?
12. New York
Justify that there is a point where f(x)= 0 on the interval [2 , 6] for the equation f(x)
f(x) = -x^2 +4x-14
13. f(x) is continuous and f(2)< 0 <f(6), therefore IVT guarantees c, 2< c <6, where f(c)=0
Find the derivative of the given function.
f(x)= (sqrtx)(-cos(x))
14.
f'(x)=(1/(2sqrtx)xx-cos(x))+(sqrtx xx sinx)
On the interval [-2,2], where does the function f(x) have a tangent line parallel to the secant line for the interval? Justify your reasoning.
f(x)=x^3-3x+2
15. f(x) is continuous and differentiable so MVT guarantees a place where the tangent and secant lines are parallel.
x = +- sqrt (4/3)
During a snow storm, snow is falling at a rate of f(t) = 1.25t + 0.5 inches per hour. Mike starts shoveling at a rate of r(t) = 0.45t + 2 inches per hour. How much snow is on the ground if Mike started shoveling at 4 inches of snow and he shoveled for 3 hours?
16. 3.1 inches
Rotate the solid whose base is bounded by f(x) and the x axis around the y=3 from the bounds x=0 and x=5.
f(x) = 1/2 x
17.
27.038 pi or 85.085
What was the last class that Ms. Braney advised?
18. Trick question, the 2023 class for their senior year and the 2019 class for all four years
lim_(x->0) sin(2x)/sin(4x)
19.
1/2
Find the average velocity of an object over the first 8 seconds it is falling, where s(t) is given in feet and t is time in seconds.
s(t)=0.63x^3-5x^2+7.8
20. 0.32 ft/s
A house owner has 3600 feet of fencing and wants to make a rectangle fence around his yard but not along the length of his house. What are the dimensions of the field that has the largest area?
21. x = 1800 feet, y = 900 feet
Given the following function with the bottom bound being 2 and the top bound being x, find F'(x):
F(x) = int cos(t^2)dt
22. cos(x)2
Rotate the solid whose base is bounded by 2y, the x-axis, and x=5 about the y-axis from the bounds y=0 and y=3.
23.
39pi or 122.522
How many senior trips has Ms. Braney chaperoned?
24. 12 senior trips
f(x) = {(x^4,+,5,,x>2),(x^2,+,13,,x<2):}
25.
lim_(x->2^+)f(x)=21
lim_(x->2^-)f(x)=17
lim_(x->2)f(x)=DNE :' LHL != RHL
A water tank has the shape of an inverted circular cone with a base radius of 3 meters and a height of 4 meters. If water is being pumped into the tank at a rate of 1 m3/min, determine the rate at which the water level is rising when the water is 2 meters deep.
26.
(dh)/dt=(4/9pi) m//min
Given f(x) on the interval [0,3], decide if Rolle's Theorem can be applied and if so where f'(x) = 0. Justify your response.
f(x)=x^4-6x^3+11x^2-6x
f(x) is both continuous and differentiable and f(0) = f(3).
x = 3, 1/2, 1
At time t = 0, x = 8. If the particle's velocity is given by the function, v(t), what is the position of the particle at t = 15?
v(t) = abs(t^3 -4^2 +5t +3)
28. 8,771.75
Determine the general solution to the differential equation below.
dy/dx=y/(2x)
29.
y = c sqrt x
What other physical classroom did Ms. Braney teach in?
30. C117, this room directly below in the English wing