34B Quick Review

Derivatives

100

ƒ(x,y) = 10x²y¹⁰

ƒ_y(x, y) = ?

ƒ_y(x,y) = 100x²y⁹

200

ƒ(x,y) = 2e^{2y - 8x}

ƒ_x(x,y) = ?

ƒ_y(x,y) = -16e^{2y - 8x}

300

ƒ(x) = 6x^ecos(x/2)

ƒ′(x) = ?

ƒ′(x) = 6ex^e-1cos(x/2) - 3x^esin(x/2)

400

ƒ(x,y) = e^{3xy - 4x}

ƒ_x(x,y) = ?

ƒ_x(x, y) = (3y - 4)e^{3xy - 4x}

500

ƒ(x, y) = ye^2xycos(y)

ƒ_y(x, y) = ?

ƒ_y(x,y) = e^2xycos(y) + 2xye^2xycos(y) - ye^2xysin(y)

Integrals

100

∫ 24x² + 24x^-2 dx

8x³ - 24x^-1 + C

200

∫ 1/t   dt

ln|t| + C

300

∫ 2e^-x/2 + π   dx

-4x^-x/2 + πx + C

400

ƒ(x) = x² + 2x + 1

Find ∫ ƒ(6t)  dt

12t³ + 6t² + t + C

500

∫ (2x⁵ + 10x³ +5)/x³   dx

(2/3)x³ + 10x - (5/2)x^-2 + C

Differential Equations

100

Find the particular solution to y′ = 2y that satisfies y(0) = 10.

y(t) = 10e^2t

200

Find the particular solution to y′ = 2(60 - y) that satisfies y(0) = 220.

y(t) = 60 + 160e^-2t

300

Find the particular solution to y′ = 5y(1 - y/100) that satisfies y(0) = 4.

y(t) = 100/(1 + 24e^-5t)

400

Find the particular solution to y′ = (1/5)(1 - y/200) that satisfies y(0) = 10.

y(t) = 200 - 190e^-t/1000

500

Find the particular solution to y′ = 0.1(5 - y) that satisfies y(10) = 50.

y(t) = 5 + (45e)e^-0.1t

Tangent Planes

100

Find the tangent plane to

ƒ(x,y) = 3x² + 10y

at the point (1, 1)

z = 6x + 10y-3

200

Find the tangent plane to

ƒ(x,y) = 5x² - y³

at the point (-1, 2)

z = -10x - 12y + 11

300

Find the tangent plane to

ƒ(x,y) = 5x²y³

at the point (-1, 2)

z = -80x + 60y - 160

400

Find the tangent plane to

ƒ(x,y) = 2xe^x-y

at the point (-2, 1)

z = -2e^-3x + 4e^-3y - 12e^-3

500

Find the tangent plane to

ƒ(x,y) = 2xye^xy

at the point (1, -1)

z = -2e^-1

Word Problem Hell

100

A radio-active element is decaying.

What is the exponential decay equation?

200

The distance a car has traveled is found from it's velocity function.

What is the Fundamental Theorem of Calculus (Part II)?

300

A cup of hot chocolate is cooling.

What is the Exponential Decay Towards a Limiting Value Equation (or Newton's law of cooling)?

400

Find the opportune time to buy/sell a stock.

What is the Second Derivative Test?

500

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