Anna's Units 5.5-5.8 Part 1
Endpoints? Define.
What is end of the interval?
What are the coordinates for Absolute Extrema (on an interval)?
What is critical points & endpoints?
Relationship that involves multiple variables
What is implicit relations?
The smallest product of two numbers given that one number is exactly 7 greater than the other number
What is p=y2+7y
Extreme Value Theorem
What is if a function f is continuous over the interval [a,b], then f has at least one minimum value and at least one maximum value on [a,b]
What are the steps to the Second Derivative Test?
What is: 1) Find the 1st & Second Deriv 2) Set the 1st deriv=0 3) Plug critical points into the 2nd deriv 4) Justify
Point of Inflection? Define.
What is a point of inflection is where the graph changes concavity?
Derivative equals zero or does not exist
What is here you would find a critical point?
dy/dx=x/y+lnx where f(1)=-2. (Max or min)
What is neither because derivative does not equal zero?
Mean Value Theorem
What is where the instantaneous rate of change equals the average rate of change over an interval where f is differentiable
Given the graph of f''(x), find the points of inflection and state the intervals of concavity: (Anna will Airdrop)
What is?
Concave Down: (-6, -4) (-2, 0) & (3, 6)
Concave Up: (-7, -6) (-4, -2) (0, 3) & (6, infin)
Points of Inflection: x=-6, -4, -2, 0, 3, & 6
Given the graph of f'(x), find the points of inflection & state the intervals of concavity: (Anna will Airdrop)
Whats is?
Concave Down: (-4, 0) (3, 5)
Concave Up: (0, 3) (5, infin)
Points of Inflection: x=0, 3, & 5
Derivative is greater than zero
What is concave up?
What is decreasing because dy/dx(-1,2)<0
The Critical Points of g(t) = 2/t^2 - 4
5 on 5.2 practice
What is t = 0
Sketch f given the graph f': (Anna will Airdrop)
What is? (Anna will Airdrop)
Find the absolute maximum value and the absolute minimum value of the function x2-2x+35 on the interval [2, 10]. Remember to show that you checked ALL the candidates.
What is?
1)f'(x)=2x-2
2)x=1
3)x=2 & x=10
4)f(1)=(1)2-2(1)+35=34 f(2)=(2)2-2(2)+35=35 f(10)=(10)2-2(10)+35=115
f(1)=abs min & f(10)=abs max
Derivative is less than zero
What is concave down?
The radius of a cylindrical soda can with volume of 512 cubic inches that will use the minimum material
What is A=r(pi)r2+1024/r
Answer to 3 on 5.1 practice A and B
A. What is according to the IVT, there is a not value, c, such that h(c) does not equal 50 and 6 less than or equal to t and t is less than or equal to 10
B. According to the MVT, there must be a value, c, where h’(c) = 3 and 10 is less than or equal to c and c is less than or equal to 40
Use the Second Derivative Test to find the relative extrema of f(x)=x3+4x2-3x+12
What is?
1) f'=3x2+8x-3 f''=6c+8
2) 3x2+8x-3=0 (3x2-x)+(9x-3)=0 x(3x-1)+3(3x- 1)=0 x=1/3 x=-3
3) f''=6x+8 f''(1/3)=6(1/3)+8=10>0 f''(-3)=6(-3)+8=-10<0
4) rel min at x=-3 b/c f'(-3)=0 & f''(-3)<0. rel min at x=1/3 b/c f'(1/3)=0 & f''(1/3)>0
Find the intervals of concavity for x2+3x-27
What is? (Anna will Airdrop)
Draw a picture, write an equation , write your equation in terms of a single value, determine max or min (find derivative), determine and verify end points
What are the steps to solving optimization problems?
A rectangle formed with the base on the x-axis and the top corners on the function y= 6-x2. (find largest possible area of each object, given its boundaries, only set up equation)
What is A=12x-2x3
Question 3 on 5.2 Practice Answer