Extreme Value
The highest y value at which the derivative of an equation equals zero.
What is the absolute maximum?
The two requirements of an interval to find a point c with Mean Value Theorem
What is continuity and differentiability?
The requirement for intermediate value to be applicable to a function on the interval from [a,b].
What is continuity over the interval [a,b]?
Requirements of the use of the First Fundamental Theorem of Calculus:
A. continuous on a closed interval
B. F(x) is the antiderivative of f(x)
C. differentiability of f(x)
What is A and B?
The lower limit of an integral where the second fundamental theorem of calculus is applicable
What is a constant?
y=-52 at x=3 for the equation: f(x)=x3-27x+2 on [0,3].
What is the absolute minimum?
The number(s) that satisfy the point c with the Mean Value Theorem for f(x) = x2-6x+12 on [0, 3]
What is c = 7/2 or 3.5?
This interval must contain a solution to g(x)=3. (g(x) is continuous.)
x = -1, 2, 4, 5
g(x) = 0, 4, 9, 15
What is [-1,2]?
Evaluate the definite integral ∫ 4x-x2 dx on the interval [0, 4].
What is the solution 32/3?
The derivative of ∫sin(t)dt from 0 to x
What is sin(x)?
The absolute maximum value of f(x)=x4-3x3-1 on [-2,2].
What is 39 at x=-2?
The number(s) that satisfy the point c with the Mean Value Theorem for h(x)=4x3-8x2+7x-2 on [2, 5]
What is c = 3.6294?
The intermediate value theorem guarantees this.
What is the existence of a solution?
Evaluate the integral ∫sinx dx from [0, π]
What is the value 2?
F'(x) given F(x)=∫(t2+2t-1)dt from -3 to x
What is x2+2x-1?
1 at x=0 for the equation f(x)=x+cosx on [0,2π].
What is the absolute minimum?
f(x) is continuous and differentiable on the interval [-7, 0] that f(-7) = 3 and f'(x) <= 2. Find the largest possible value of f(0)
What is f(0) <= 11?
On someone's first day of college they weighed 160 lbs. By the end of their freshman year they weighed 175 lbs. Which weights they absolutely weigh at some point during their freshman year?
A. 159 lbs
B. 164 lbs
C. 185 lbs
D. 172 lbs
What is B and D?
Evaluate the integral ∫(x+√x)/4 dx on the interval [0, 1]. Find the solution
What is the value 7/24 or 0.292?
∫(-2t+3)dt from -5 to x2
What is -4x3+6x?
The absolute minimum value of f(x)=sinx on [0,2π].
What is -1 at x=3π/2?
The number(s) that satisfy the point c with the Mean Value Theorem for A(t) = 8t + e-3t on [-2, 3]
What is c = -1.0973?
h is a continuous function on the closed interval [-3,4]. h(3)=-1 and h(1)=7. These options of the ones below are guaranteed by the intermediate value theorem.
A. h(c)=1 for at least one c on [-3,4]
B. h(c)=-2 for at least one c on [-1,2]
C. h(c)=1 for at least one c on [1,2]
D. h(c)=5 for at least one c on [-3,4]
What is A and D?
A ball is thrown straight up with velocity given by v(t)=−32t+20 ft/s, where t is measured in seconds. Find and interpret ∫v(t)dt from [1, 0].
What is the value 4?
F'(2) if F(x)=∫dt/t2 from 3 to x2
What is 1/4?