Theorems
If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.
What is the Extreme Value Theorem?
If f(x)=x1/2+(3/x1/2) then f’(4)=
1/16
What is the Integral of (5e2x+(1/x))?
(5/2)e2x+lnlxl+C
What is the limit as x approaches -3 of ((x2-9)/(x2-2x-15))
3/4
What is the area of the region bounded by x=y2-4y and x=2y-y2?
9
If a function f(x)is continuous on a closed interval [a,b], and has is differentiable on the open interval (a,b). Then there must be at least one value x in the open interval (a,b), call it c, where the tangent line is parallel to line AB. This is the slope of the tangent line to the graph of f(x)at x=c is equal to the slope of the line AB.
What is the Mean Value Theorem?
If f(x)=x2-4 and g is a differentiable function of x, what is the derivative of f(g(x))?
2g(x)g’(x)
Which of the following is an antiderivative of 3sec2(x)+2
3tan(x)+2x
What is the limit as x approaches infinity of ((10–6x2)/(5+3ex))
0
What is the volume of the solid generated by revolving the region bounded by y=x3, the y-axis, and the line y=27 about the y-axis?
Pi (integral from 0 to 27 of y1/3)dy
If a function f(x) is continuous on a closed interval [a,b]. Then the function takes on every value between f(a) and f(b).
What is the Intermediate Value Theorem?
If y=sin(x)cos(x), then at x=(pi/3), (dy/dx)=
-(1/2)
What is the Integral of (x2(x3+5)6)
(1/21)(x3+5)7+C
What is the limit as x approaches zero of ((sin(x))/(ex-1))
1
Let S be the region enclosed by the graphs of y=2x and y=2x2 for 0<x<1. What is the volume of the solid generated when S is revolved about the line y=3?
Pi (Integral from 0 to 1[(3-2x2)2-((3-2x)2)]dx
If a function is continuous on a closed interval [a,b], and has a derivative on the open interval (a,b), and f(a)=f(b), has the same y-value at the endpoints, a and b. Then there must be at least one value of x, c, in the open interval (a,b) where the function has a horizontal tangent, f’(c)=0.
What is Rolles Theorem?
If y3+y=x2, then (dy/dx)=
2x/(1+3y2)
If (dy/dt)=-10e-t/2 and y(0)=20, what is the value of y(6)?
20e-3
What is the limit as x approaches zero of (((4+x)1/2-2)/(2x))
1/8
The slope of the tangent to the curve y3x+y2x2=6 at (2,1) is
-5/14
Let f be a continuous function on the closed interval [-3,6]. If f (-3)=-1 and f(6)=3, then the Intermediate Value Theorem guarantees that
-1<f(x)<3 for all x between -3 and 6.
If (dy/dx)=2-y, and if y=1 when x=1, then y=
2-e1-x
If f’(x)=cos(x2) and f(3)=7, then f(2)=
6.759
What is the limit as x approaches -2 of ((x3-x2-10–8)/(5x3+12x2-2x-12))
3/5
Let R be the region enclosed by the graphs of g(x)=-2+3cos((pi/2)x) and h(x)=6-2(x-1)2, the y-axis, and the vertical line x=2
Write but do not evaluate an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y=6.
Pi (integral from 0 to 2 (6-(g(x)))2 - (6-(h(x)))2)dx