Limits
The lim x-->2 (8 - 3x + 12x^2) is...
= 8 - 3(2) + 12(4)
= 50
The derivative of 2x^4 + 5x^3+ 3x^2 + 3x + 4 is...
= 8x^3 +15x^2 + 6x + 3
The general solution to the integral ∫ (4x^6 - 2x^3 + 7x - 4) dx is...
= (4x^7)/7 - (1x^4)/2 + (7x^2)/2 - 4x + C
The left Riemann sum of the function f(x) = x^2 from interval 0 to 3 with three equal subintervals is...
= (1)(0) + (1)(3) + (1)(9)
= 12
The dy/dx of the equation 4x^2 - y^2 = 9 is...
= 4x/y
The lim x-->-3 ((6+4x)/(x^2 + 1)) is...
= 6+4(-3)/(-3)^2 + 1
= -6/10
= -3/5
The derivative of cos(3 sin(3x)) is...
= -sin (3 sin(3x)) * (3 cos (3x)) * 3
= -9 sin(3sin(3x)) * cos(3x)
The general solution to the integral ∫ (z^7 - 48z^11 - 5z^16) dz is...
= (1z^8)/8 - 4z^12 - (5z^17)/17 + C
The midpoint Riemann sum of the function f(x) = 3x^2 + 4 from interval 0 to 6 with three equal subintervals is...
= 2(7) + 2(31) + 2(79)
= 14 + 62 + 158
= 234
The dy/dx of the equation 1/x + 1/y = -1/12 is...
= -y^2/x^2
The limx-->-5 ((x^2 - 25) / (x^2 + 2x - 15)) is...
= (x-5)(x+5)/(x-3)(x+5)
= (x-5)/(x-3)
= 5/4
The derivative of (x+3)(5x^3+3) is...
= (x+3)(15x^2) + (1)(5x^3 + 3)
= 15x^3 + 45x^2 + 5x^3 + 3
= 20x^3 + 45x^2 + 3
The general solution to the integral ∫ (10t^-3 + 12t^-9 + 4t^3) dt is...
= -5/t^2 - 3/2t^8 + t^4 + C
The right Riemann sum of the function f(x) = 6x^2 + 4 from interval 0 to 3 with three equal subintervals is...
= (1)(10) + (1)(28) + (1)(58)
= 96
The dy/dx of the equation x^2 + xy + y^2 = 10 is...
= (-2x - y) / (x + 2y)
The limx-->8 ((2x^2 - 17x + 8) / (8 - x)) is...
= (2x - 1)(x - 8) / -(x - 8)
= (2x - 1) / -1
= -15
The derivative of (4x^3 - 3)/(5x^2) is...
= [(5x^2)(12x^2) - (10x)(4x^3 - 3)]/ (5x^2)^2
= [60x^4 - 40x^4 + 30x] / 25x^4
= [20x^4 + 30x] / 25x^4
The general solution to the integral ∫ (4e^x + 15 - 1/6z) dz is...
= 4e^x + 15z - 1 ln |z|/6 + C
The trapezoidal sum of the function f(x) = 3x^2 + 4 from interval 0 to 6 with three equal subintervals is...
= (1/2)(2)(4+16) + (1/2)(2)(16+52) + (1/2)(2)(52+112)
= 20 + 68 + 164
= 252
The dy/dx at point (1,1) of the equation x^6 + y^6 + 6(x^2)(y^2) - 8 = 0 is...
= -1
The limx-->4 ((x^1/2 - 2) / (x - 4)) is...
Use L'Hospital
= (1/2x^-1/2) / 1
= (1/2) / (x^1/2)
= 1/4
The fourth derivative of 3x^7 - 6x^4 + 8x^3 - 12x + 18 is...
=21x^6 - 24x^3 + 24x^2 - 12
=126x^5 - 72x^2 + 48x
=630x^4 - 144x + 48
=2520x^3 - 144
The general solution to the integral ∫ (6cos(z) + 4/(1-z^2)^1/2) dz is...
= 6sin(z) + 4sin^-1(z) + C
Remember* ∫ du/(a^2-u^2)^1/2 = arcsin u/a + C
The trapezoidal sum of the function f(x) = 3x^2 + 4 from interval 0 to 9 with three equal subintervals is...
= (1/2)(3)(4+31) + (1/2)(3)(31+112) + (1/2)(3)(112 + 247)
= 52.5 + 214.5 + 538.5
= 805.5
The dy/dx at point (2,-4) of the equation 2x^2 - 3y^2 = 5xy is...
= -2