Everything Limits
Those Crazy Derivatives
Integration
Rectangle Approximation
Related Rates & Optimization
100
is the function {5x+7, x<3 {7x+1, x > or equal to 3 continuous at x=3?
YES
100
What causes a derivatives not to exist?
hole asymptote jump corner endpoint
100
What two things does the fundamental theorem of Calculus tell us?
1.) When you have an integral with bounds, they can be applied to the function F(b)-F(a) 2.) If the top bound is a variable, then the derivative of the integral is the original function in terms of that variable
100
What is the Riemann's Sum for Right Rectangle Alignment Method?
An=(b-a)/n * the sum from i=1 to n of f(xi) or An=delta x * f(x1) + f(x2) + f(xn)
100
Steps for solving optimization problems?
1.) draw diagram 2.) collect given info and equations 3.) establish what we are trying to optimize 4.) write the function to be optimized in terms of one variable 5.) use first derivative to find the max/min 6.) verify with number line
200
Limit as x approaches 3 of (x^2-2x-3) / (x-3)
4
200
derivative of 2sin(3x)cos(4x)
6cos(3x)cos(4x) - 8sin(3x)sin(4x)
200
the integral of (x^2)sec^2(x^3) dx
1/3 tan (x^3) + c
200
Find the area under the curve y=2x-x^2 from x=1 to x=2 with n=4 using MRAM
43/64 or .67
200
Steps for related rates problems?
1.) draw diagram label given info 2.) identify known and needed rate of change 3.) find equation that relates variables in problem 4.) differentiate both sides with respect to time 5.) plug in known values and evaluate
300
limit as x approaches infinity of (x^4-8) / (10x^4+25x+1)
1/10 or .1
300
Name the derivatives of the following functions: a^x tanx log base a of x inverse cotx cscx
a^xln(a) sec^2 x 1 / (xln(a)) -1 / (1+x^2) -cotxcscx
300
integral of dx/(x-1)^2
-1 / (x+1) + c
300
given the chart, using LRAM with n=5 to approximate the integral x 0 1 3 5 9 14 g(x) 10 8 11 17 20 23
216
300
A spherical balloon is inflating at a rate of 27pi in^3/sexc. How fast is the radius of the balloon increasing when the radius is 3in?
3/4 in/s
400
limit as h approaches 0 of [(1/x+h) - (1/x)] / h
1/x^2
400
derivative of (1+cot(2/x))^-2
[(4/x^2)(csc^2(2/x))] / (1+cot(2/x))^3
400
find the derivative of the integral (t^2 - t) dt on the interval from 5 to 3x
27x^2 - 9x
400
use RRAM to approximate the integral of cosxdx from -pi/2 to pi/2
around 2 units^2, this is exact answer
400
An open top box is made by cutting congruent squares of side length x from the corners of a 18 by 24 inch sheet of cardboard and bending the sides up. How large should the squares be to make the bx hold as much as possible? *if needed view diagram on test*
L = 17.22 w = 11.22 h = 3.39
500
limit as x approaches infinity of [e^3x - e^5x] / x
-2
500
equation of the line tangent to the graph of y= (x^2+4) / (x-6) at x = 5
y+29 = -39(x-5)
500
find the area between the curve y=x^2-4x-5 and the curve y=2x-5
36 units^2
500
What is the Riemann's Sum for MRAM
An= delta x * the sum from i=0 to n-1 of f((xi+xi+1)/2)
500
use the first and second derivative tests to describe all behavior of the unction f(x)=x^3 - 6x^2 +9x +1. be sure to include intervals where it is increasing, concave up, etc.
Critical Values: 3,1,2 increasing, decreasing, decreasing, increasing concave d, concave d, concave up, concave up
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