Limits & Continuity
What is the formula for instantaneous velocity?
f(a+h)-f(a)/h
What is the definition of a derivative?
the slope of a tangent line at a point/instantaneous velocity
What is the difference between an "absolute maximum" and a "relative maximum"?
Absolute maximum values are the highest values in the function that enclose the interval (only exist on a closed interval). Relative maximum values exist on an open or closed interval (the curve changes from increasing to decreasing on the interval).
Can you find the same value (of area under a curve) using LRAM/RRAM and MRAM? Will MRAM always be bigger?
MRAM will not always be bigger. It is possible to come across a case where the LRAM/RRAM and MRAM values are equal.
Find limx->0
y=x+4 [-4,0]
y=x-3 (0, 4]
The limit does not exist.
What is the limit of a function that has a larger coefficient on the bottom? (x^2/x^3)
zero (0)
What is the derivative of f(x)?
f(x)= x^4-2x^2+5
4x^3-4x
What is the Mean Value Theorem and what are the requirements?
(f(b)-f(a))/(b-a)
continuous and differentiable at every point on the closed interval
Solve the integral:
[-2,4] (x/2 + 3)dx
1/4x^2 + 3x [-2,4]
answer=21
what is f'(x)?
f(x)= e^(2x/3)
f'(x)= 2/3* e^(2x/3)
What is the limit of this function: limx-->3 (x^2-x-6)/(x-3)
a)-1 b)1 c)5 d)does not exist
c) 5
What process is needed to find f'(x)? FIND f'(x)!
sin^2(2x)
chain rule, 4cos(2x)
Solve using MVT:
f(x)=arcsinx [-1,1]
pi/2
Suppose that f and h are continuous functions and that...
[1,9] f(x)dx=-1
[7,9] f(x)dx=5
[7,9] h(x)dx=4
find a) [7,9] [f(x)+ h(x)]dx
b) [7,9] [h(x)-f(x)]dx
a) 9
b) 1
Let f be the function defined as
f(x)= {
3-x, x<1
ax^2 + bx, x>=1
a and b are constants. Find the values of a and b that make f(x) differentiable and continuous.
b= 5
What value of k makes f(x) continuous?
f(x)= {1/3x+6, x<9 & {k-2x, x>=9
k=27
find f'(x)
f(x)= cosx/sinx
f'(x)= -csc^2x
f(x)= 4x^3+21x^2+36x-20
[-7/4, infinity]
Graph these equations on your calculator and find the area between the curves:
y=2x^2
y=x^4 - 2x^2
128/15
The region enclosed by the x-axis and the arch of the curve y=2sinx and each cross-section is cut perpendicular to the x-axis as a semicircle whose diameter runs from the x-axis to the curve.
find the volume.
volume= 2.467in^3
What is the limit?
limx-->0 (ln(1-x)-sinx)/(1-cos^2x)
The limit does not exist!!
Find the derivative:
(sinu/cscu) + (cosu/secu)
zero (0)
A piece of cardboard measures 10x15in. 2 equal squares are removed from the corners of the 10in side. 2 equal rectangles are removed from the other corners.
Write a formula for the volume of the box AND find the maximum volume given the measure of x.
volume= 132.04
x=1.96
The region in the first quadrant is enclosed by the y-axis and the graphs of y=cosx and y=sinx is revolved around the x-axis to form a solid. Find the volume.
volume= pi/2
Who are the best calc students who created this Jeopardy board?
Mollie and Rena