MATH 151A Jeopardy

Limit

100

Find lim x->2 [(x^2-x-2)/(x-2)]

lim x->2 (x+1)=3

200

Find lim t->0 [(t^2+9)^1/2 -3]/(t^2)

1/6

300

Evaluate lim x->infinity (3x^7-4x^2+1)/(5-10x^2)

-infinity

400

On what intervals is g(x)=(x^2+2x+7)/(x^2-1) continuous?

(-infinity, -1) U (-1, 1) U (1, infinity)

500

Evaluate lim x->4 (2x^3-128)/(x^(1/2)-2)

384

Derivatives

100

Find the derivative of the function f(x)=x^2-8x+9 using the limit definition of derivative.

f'(x)=2x-8

200

Find the derivative of y=(x^2)(2x) (USE PRODUCT RULE)

y'=6x^2

300

Find the d/dx (4x^2-x)(x^3-8x^2+12)

20x^4-132x^3+24x^2+96x-12

400

Find the derivative of f(w)=(3w+w^4)/(2w^2+1)

f'(w)=(4w^5+4w^3-6w^2+3)/(2w^2+1)^2

500

Differentiate y=(2x+1)^5[(x^3-x+1)^4]

y'=4(2x+1)^5(x^3-x+1)^3(3x^2-1)+10(x^3-x+1)^4(2x+1)^4

Trig

100

What is cos(pi/3)

1/2

200

Evaluate lim x->0 (sin (2x))/5x

2/5

300

Find the derivative of y=-sin (2x)

y'=-2 cos (2x)

400

Find d/dx [-2 sin^2 (2x)]

-8 cos (2x) sin (2x)

500

Differentiate f(x)=(sec x)/(1+tan x)

f'(x)=[sec x (tan x -1)]/(1+tan x)^2

Implicit Differentiation

100

Find y' if x^4+y^7=1

y'=-(4x^3)/(7y^6)

200

Find y' if x^3+y^3=6xy

y'=(2y-x^2)/(y^2-2x)

300

Find dy/dx if x^3+4xy+y^2=13. Evaluate dy/dx at the point (1, 2)

dy/dx = -11/8

400

Find the points where the tangent is horizontal: x^2+xy+y^2=4

(2/(3)^1/2, -4/(3)^1/2), (-2/(3)^1/2, 4/(3)^1/2)

500

Find y' if sin (x+y)=y^2 cos (x)

y'=(y^2 sin (x)+cos (x+y)) / (2y cos (x)-cos (x+y))

Derivatives with Trig

100

d/dx [x^3(tan (x)) (3x^2-9)]

3x^2(tan (x)) (3x^2-9)+x^3 (sec^2 (x)) (3x^2-9)+x^3(tan (x)) (6x)

200

Find the derivative of function y=cos(a^4+x^4)

y'=-4x^3 (sin (a^4+x^4))

300

Find the second derivative of y=cos (x^2)

y''=-4x^2 cos (x^2)-2 sin (x^2)

400

Find the derivative of function y=sin (sin (sin (x)))

y'=cos (sin (sin (x))) cos (sin (x)) cos(x)

500

Find the derivative of the function y=cot^2(sin x)

y'=-2 cos (x) cot(sin (x)) csc^2(sin (x))