Extreme Values
What is an absolute extreme value?
The greatest or smallest value of f(x) on the function
What is the formula for the Mean Value Theorem?
f'(c)=(f(b)-f(a))/(b-a)
What is the limit definition of a Derivative?
f'(x) = limit as h approaches 0 of (f(x+h)-f(x))/h
When is the function increasing? f(x) = x2+6x+24
Increasing (-3,∞)
What is the derivative of a2+b2=c2 in respect to time?
2a(da/dt)+2b(db/dt)=2c(dc/dt)
What is the relative minimum of f(x)=x2
What is y=0
When does a given function satisfy the conditions of the Mean Value Theorem?
When the function is continuous over [a,b] and differentiable over (a,b).
What is Derivative of f(x) = x2-4x+2
f'(x) = 2x-4
When is the function Increasing and Decreasing? f(x)= 2x3+3x2-12x
Increasing (-∞,-2), (1,∞) and Decreasing (-2,1)
What is the derivative of the circumference of a circle in respect to time?
dC/dt=2π(dr/dt)
How many critical points does the function f(x) = x3-5x2+2x+8 have?
True or False: You can apply the MVT on the interval [-3,3] in the function f(x)=1/(x2-x-2)
False: There are infinite discontinuities on x=2 and x=-1
What is the Derivative of f(x) = 5xsin(x)
What is f'(x)=5(sin(x)+xcos(x))
What is the concavity of this function? f(x)=x3+9x2+8x
What is Concave up on (-3,∞) and Concave down on (-∞,-3)
What is the derivative of the volume of a cube in respect to time?
dv/dt=3x2(dx/dt)
Find the relative extrema of the function f(x) = x/(x2+1). Where do these values occur?
Relative Max: (1,1/2) Relative Min: (-1,-1/2)
Find a value of C that satisfies the Mean Value Theorem on f(x) = sin(x) [0,2π]
C = π/2, C = 3π/2
What is the Derivative of f(x) = sin(x)/(sin2(x)+cos2(x))
f'(x) = cos(x)
What is the concavity and point of inflection of f(x)=x3-6x2+4
Concave Upward (-∞,2) and Concave Downward (2,∞). Point of Inflection on (2,-12)
What is the derivative of the volume of a cylinder in respect to time?
dv/dt=2πr(dr/dt)h+πr2(dh/dt)
What are the extreme values of V(x) = x(10-2x)(16-2x) over the domain 0 < x < 5
Absolute Max at the point (2,144)
Ms. Mountain takes 1 second to run to The Great Tea and Crumpet Cafe which is 5 miles from her house, she then realizes she’s late for school which is 50 miles from her house. She needs to run there in 4 seconds or she will be late and tea will become illegal. What is her average rate of change and is there a value c that satisfies the MVT.
10 miles/sec No value of c satisfies the MVT
What is the derivative of the function f(x)=cos2(4x)+sin2(4x)
f'(x)=0
What are the Infection points, Concavities and Extremas of this function? f(x)=x4-4x3+2
What is Inflection Points on (0,2) and (2,-14), concave upward (-∞,0) and (2,∞). Concave Downward (0,2) and Relative Minimum on y=-25.
What is the derivative of x2+7=cos(y) in respect to x?
what is (dy/dx)=(2x)/-sin(y)