Limits
f/f'/f''
Integrals and application
Derivatives
Theorems
100
The limit of 3x/2+x as x approaches 1
What is 1?
100
F''(x)>0 and F'(x) is increasing
What is concave up?
100
Integral of (x^2)+2
What is ((x^3)/3)+2x+C?
100
Find the derivative of 5x^3+8x
What is 15x^2+8?
100
Integral from 8 to 8 of 7x^3+e^2x+8
What is 0?
200
The limit of (x^2)+25x/(20x^3) as x approaches 10
What is 0.0175 (7/400)?
200
F(-2)
What is ~4?
200
Integral from -2 to 2 of 3x
What is 0?
200
f(x)= (8x^4)+(12x^3)+(4x^2)+(12x)+1000
Find f''(x)
What is f''(x)= (96x^2)+72x+8?
200
If f is continuous on [-1,1], f(-1)=7, and f(1)= -5, then there is a zero between -1 and 1. (True or False)
What is true?
300
The limit of (3^(x)+8(x^2)+5)/(8x^(3)+9x^(2)+1) as x approaches negative infinity
What is 0?
(1/very big number)
300
Critical point(s) of f(x)
What are x=-2 and x=0?
300
Find the area bounded by y=sqrt(x), x= 9, and the x-axis
What is 18?
300
Derivative of 6x*(sqrt(x))
What is 9x^(1/2)?
300
x 1 2 3 4 5 6
f(x) 2 6 7 3 8 1
Use MVT to determine if there exists a value where f'(c)= 5.
What is yes (between f(5) and f(4))?
400
The limit of ((10x^3)+(2x^2)+3)/((3x^3)+(6x^2)) as x approaches infinity
What is 3.3333 (10/3)?
400
F''(-2)
Hint: f'(x)= 3x^2+6x
What is a negative value since f(x) is CCD and f'(x) is decreasing (without hint)?
What is -6 (with hint)? Find the derivative and then solve by plugging in -2
400
Integral of ((ln(x))/x)dx
What is 0.5((ln(x))^2)+C?
400
Given the 10e^x +8x^2 = 10x+7ln(x), use derivatives to show that 10e^x+16x-(7/x)-10=0
10e^x+8x^2= 10x+7ln(x)
10e^x+16x= 10+(7/x)
10e^x+16x-(7/x)-10= 0
400
If f(x)= sin(x/3)+10 then there exists a number "z" in the interval (pi/2)<x<(3pi/2) that satisfies the conclusion of the Mean Value Theorem. Find z.
What is about 344.6 degrees (or 6.205 radians)?
Work:
f'(z)= (1/3)(cos(x/3))
f'(z)= (f(3pi/2)-f(pi/2))/ pi
(sin(3pi/2)-sin(pi/6))/ pi
1/3cos(x/3) = -1/2pi then solve for x
500
The limit of ((x^2)-4)/(x^(3)+7X-22)
What is 4/19?
500
Given f(x)=(x+2)(x-3)^3, find the value(s) of x for which f has a local maximum and minimum.
What is a local minimum at x= -3/4?
500
Integral of ((x^2)+6x-8)/(sqrt(x)) dx
What is ((2/5)x^(5/2))+4x^(3/2)- 16x^(1/2)+C?
500
Let f(x)=1/2(x^2)+8x-23 and g(x) be the inverse of f(x). Given f(-4)=14, find g'(14).
What is 1/4?
500
Let g(x)=(x^4)+(8x^3)-(2x^2)+1. What is the minimum value of g on [-7,1]?
What is -6-(sqrt(40))/2?