Derivatives
Riemann Sum
Integrals
Differential Equations
Limits
100
Find the derivative of the equation f(x)= 6x+5x+3
f ’(x)= 6x+5
100
F(x) = 2x + 3 [0,4] n= 4 LRAM
Width: (4-0)/4= 1 LRAM= 1[f(0) = f(1) + f(2) + f(3)] LRAM= [ 3 + 5 + 7 + 9] LRAM= 24 units
100
{(cosx)/((sinx)-3) dx
= ln(sin(x)-3) + C
100
A 98 degree cake is put into a 14 degree freezer. Given the equation T=ce^kt+18, find “c”.
80
100
Lim x^2 -64/ x-> 8 x- 8
Lim (x-8)(x+8)/ -> lim (x+8)= 8+8= 16 (x-8) x-> 8
200
Find the derivative of f(x)= 10xy
f ’(x)= 10x+10y
200
F(x)= x^2 + 2 [0,6] n= 3 MRAM
Width: (6-0)/3 = 2 MRAM= 2[f(1) + f(2.5) + f(4.5)] MRAM= 2[ 3 + 6.25 + 20.25] MRAM= 59 units
200
{xcot(3x^2)dx
= 1/6 ln(sin(3x^2)) + C
200
Given dv/dt=1000+.10V, find the equation for “V” in terms of “t”.
V=ce^.10t-10,000
200
Lim x+2/ X-> -2 x^3+ 8
0 -> (x+2)/ = (x+2)/ = 1/ 0 (x+2)(x^2-2x+4) (x+2)(x^2-2x+4) 12
300
Find the derivative of the following equation: f(x)= (8x+5)^3
f ’(x)= 24(8x+5)^2
300
F(x)= 3x^2 + 5 [1,4] n= 3 LRAM and RRAM
Width: (4-1)/3= 1 LRAM= (1)[ f(1) + f(2) + f(3)] LRAM= [8+ 17+ 32] LRAM= 57 units RRAM= (1)[ f(2) + f(3) + f(4)] RRAM= [17 + 32 + 53] RRAM= 102 units
300
{ ((2x+2)/(x^2 +2x-4)^4) dx
{ ((2x+2)/(x^2 +2x-4)^4) dx = { (2x+2)(x^2 +2x-4)^-4 dx = -1/3 (x^2 +2x-4)^-3 +C
300
What is the equation for the law of exponential growth?
B=ce^kt
300
Lim 1-cosx/ X->0 3x^2-2x^3
Lim 1-cosx/ = lim 1-cosx/ = 0 X->0 3x^2-2x^3 x->0 x(3x-2x^2)
400
Find the derivative of f(x)= x^2(cosx)
f ’(x)= 2xcosx-x^2sinx
400
F(x)= 1+x^2 [-1, 2] n= 6 MRAM
Width: (2+1)/6= 1/2 MRAM= (1/2)[f(-3/4) + f(-1/4) + f(1/4) + f(3/4) + f(5/4) + f(7/4)] MRAM= (1/2)[ (25/16) + (17/16) + (25/16) + (41/16) + (65/16)] MRAM= (1/2)(190/16) MRAM= 95/16 units
400
{ x^(1/2) (x^(1/3) - x^(2/5)) dx
{ x^(1/2) (x^(1/3) - x^(2/5)) dx = {x^(11/6) - x^(19/10) dx = 6/11 x^(11/6) - 10/9 x^(19/10) + C
400
A pie with an internal temperature of 220 degrees is placed in a 40 degree refrigerator. After 15 minutes, the temperature is 180 degrees. How long will it take to decrease to 100 degrees?
t= 65.845 minutes
400
Lim (x^2-2x)/ X-> 2^- (x^2-4x+4)
Lim (x^2-2x)/ = 0/ = x(x-2)/ = 0/ = V.A. X-> 2^- (x^2-4x+4) 0 (x-2)(x-2) 0 -> 1.9/ = -∞ 1.9-2
500
f(x)= sinx^3 tan(4x)
f ’(x)= 3(sinx)^2 cos(x) tan(4x) + 4sin(x)^3 sec(4x)^2
500
F(x)= x^2 [0,4] n= 5 TRAM
Width= (4-0)/5= 4/5 TRAM= (1/2)(4/5)[f(0) + 2f(4/5) + 2f(8/5) + 2f(12/5) + 2f(16/5) + f(20/5)] TRAM= (4/10)[ 0+ 2(16/25) + 2(64/25) + 2(144/25) + 2(256/25) + (400/25)] TRAM= (2/5)[ (32/25) + (128/25) + (288/25) + (512/25) + (400/24)] TRAM= (2/5)(1360/25) TRAM= 544/25 ~ 21.6 units
500
{ (ln^3 (x))/x dx
= 1/4 ln^4 (x) + C
500
Using the law of exponential growth equation and the coordinates (2,100) and (4,300) find what “k” is equal to.
k= .549
500
Lim ((1/4) + (1/x))/(4+x) X-> -4
Lim (1/4) + (1/x)/ *4x = x+4/ = 1/ = -1/ X-> -4 (4+x) *4x 4x(x+4) 4x 16
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