MATH 1297: Calculus I Review Jeopardy

Derivatives: Power Rule

100

d/dx (x^7)

7x^6

200

d/dx (1/3 x^12 + 1/5x^20)

4x^11 + 4x^19

300

d/dx (3/7x^14 - 3x^8-2x^-3)

6x^13 - 24x^7 + 6x^-4

or

6x^13 - 24x^7 + 6/x^4

400

d/dx (1/8x^(8/3) - 5/2 x^(12/25))

1/3 x^(5/3) - 6/5 x^(-13/25)

500

d/dx (1/x - 2/x^2 + 3/x^3 - 4/x^4 + 5/x^5)

-x^-2 + 4x^-3 -9x^-4+16x^-5 -25x^-6

Derivatives: Product Rule

100

d/dx (x^2sinx)

x^2cosx + 2xsinx

200

d/dx (sinxcosx)

-sin^2x+cos^2x

300

d/dx((3x^5-1)(x^3+5x^2+2))

(3x^5-1)(3x^2+10x) + (15x^4)(x^3+5x^2+2)

or expanded form:

9x^7 + 30x^6-3x^2-10x + 15x^7 + 75x^6 + 30x^4 = 24x^7 + 105x^6 + 30x^4 - 3x^2 - 10x

400

d/dx((5x^5-x^7)(20x^2+3x^-7))

(5x^5-x^7)(40x-21x^-8)+(25x^4-7x^6)(20x^2+3x^-7)

or expanded form:

200x^6 - 105x^-3 -40x^8 +21x^-1 + 500x^6 + 75x^-3 -140x^8 -21x^-1 = 700x^6 - 30x^-3 -180x^8

500

d/dx ((arctanx)(lnx))

(arctanx)(1/x) + (1/(x^2+1))(lnx) = arctanx/x + lnx/(x^2+1)

Derivatives: Quotient Rule

100

d/dx ((3x^2+9)/(x-5))

((x-5)(6x)-(3x^2+9)(1))/(x-5)^2 = (6x^2-30x-3x^2-9)/((x-5)^2) = (3x^2-30x-9)/((x-5)^2)

200

d/dx ((4sqrt(x))/(x^2-2))

((x^2-2)*2x^(-1/2) - 4x^(1/2)(2x))/(x^2-2)^2

=

(2x^(-1/2)(x^2-2) - 8x^(3/2))/((x^2-2)^2)

300

d/dx ((6root(3)(x))/(4x^2+1))

((4x^2+1)(2x^(-2/3))-6x^(1/3)(8x))/(4x^2+1)^2

400

d/dx ((4sinx)/(2x+cosx))

((2x+cosx)(4cosx)-(4sinx)(2-sinx))/(2x+cosx)^2

500

d/dx((1-lnx)/(x^2-lnx))

((x^2-lnx)(-1/x)-(1-lnx)(2x-1/x))/(x^2-lnx)^2

Derivatives: Chain Rule

100

d/dx ((6x^2+7x)^4)

4(6x^2+7x)^3*(12x+7)

200

d/dx (e^(1-cosx))

e^(1-cosx)*(sinx)

300

d/dx (ln(1-5x^3-x^7))

1/(1-5x^3-x^7)*(-15x^2-7x^6)

400

d/dx x^2ln(x^5)

x^2(1/x^5*5x^4) + 2xln(x^5)

500

d/dx (2sin(3x+tanx))

2cos(3x+tan(x))*(3+sec^2x)

Integrals: U-Substitution

100

\int sqrt(4x+9)dx

1/6(4x+9)^(3/2) + C

200

\int x^4/(1+x^5)^(1/3) dx

3/10(1+x^5)^(2/3)+C

300

\int cos(5x)/e^sin(5x) dx

-1/5e^-sin(5x)+C

400

\int sin(lnx)/x dx

-cos(lnx)+C

500

\int sqrt(4-sqrt(x))dx

-16/3(4-sqrt(x))^(3/2)+4/5(4-sqrt(x))^(5/2) + C