Derivatives
Integrals
Related Rates/Optimization
Natural Disasters
Miscellaneous
100
d/dx ((tan(x+2))^1/2)(cosx)
1/2tan(x+2)^-1/2*(sec(x+2))^2 - ((tan(x+2))^1/2)(sinx) Chain + Product
100
Take the integral of x^2*(x^3-1)^4
(x^3-1)/15
100
Optimize the area of a rectangle with a perimeter of 200
xy=200 2x+2y=200 x=100-y d/dx 100y-y^2=200 100-2y=0 2y=100 y=50 if y=50, then x=50, so x=y, which is a square
100
(e^x)(xtany)(ln1)(arcsin(a^7)) ------------------------------------ (((((z^7)+3x) -cosx) -1) + pi)
0
100
You want to find the height of a rocket. You know the angle at the top of the rocket's flight and the distance you were from the launch point. What trig function would you use to find the height?
Law of Sines
200
Take the derivative of (x^2)/6+(y^2)/14
dy/dx=-7x/3y
200
/2 | x^2 lnx dx /1
1.0706 units squared
200
What are critical points and inflection points?
Critical point is when the speed of the function is zero. Inflection point is where the slope changes from concave up to concave down or vice versa.
200
(x^1/7*cos) y= -------------- (x^3+x)^1/3 Hint: Use ln
dy (x^1/7*cos) / 1 3x^2+1 \ -- = -------------- *| --- + tanx - ----------- | dx (x^3+x)^1/3 \ 7x 2(x^3+x /
200
cos(arcsin(2sqrt(2))) sin(arccos(sqrt(3)/2)) arcsin(cos(60degrees))
2sqrt(2) 1/2 30 degrees
300
Do the limit method way of: f(x)=1/(sqrt(x+1)) at x=3
Equation: -1/(2(x+1)^3/2) Answer: -1/16
300
Use the limit summation method to solve 7x + x^3 from [1,4]
116.25 units squared
300
There is a pyramid with a square base with a side length of 51 m. The height, but not the base, is growing at a rate of 5 m/s. How is the volume changing in time?
dV --- = 4335 m/s dt
300
/x^7* tanx \ Expand ln |------------- | \ cosx /
7lnx + lntanx -lncosx
300
take the integral of cos(1/x^3) from 1 to infinity
-.0087262032
400
Take the double derivative of sqrt(cos(1/x)) Hint: Don't simplify the very last step
(1/4(cos(1/x)^-3/2)*-1/x^2(sin(1/x)))(-2/x^3(sin(1/x))+-1/x^4(cos(1/x))+(1/2cos(1/x)^-1/2)(-2/x^5(cos(1/x))+6/x^4(sin(1/x))+1/x^6(sin(1/x))+4/x^5(cos(1/x)))
400
Do the limit summation of an integral for: f(x)=x^2+2 (0,1)
2
400
A sphere is growing by 800 cubic centimeters per second. How is the radius changing in time? 4 V= --- * pi * r^3 3
dr 800 --- = ---------- dt 4*pi*r^2
400
take the integral of: (1-e^-x)/(e^-x+x)
ln(e^-x+x)
500
The derivative of ((x-3)/(x^2+1))^2
(-2x^3+18x^2+-34x+-6)/((x^2+1)^3)
500
/ | (x^3)(sin(x)) dx /
(3x^2 - 6)sinx-(x^3-6x)cosx+C
500
Simplify this!! 1/3[5/2(4/7lnx-5/8lnx)]
What is ln[(x^-3/56)^5/6]
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