Ace it!

Domain and Range

100

Find the domain of: 

x²  + 9x + 13 > -7 

(-∞, -5) ∪ (-4, ∞)

200

Find the domain of: 

x² - 5x -14 / x + 1 > 0 

(-2, -1) ∪ (7, ∞)

300

Find the domain of: 

y = (2x+5) / (x² - 3x + 2) 

(-∞, 1) ∪ (1,2) ∪ (2, ∞)

400

Find the domain of: 

y= √(x² - 49) / (x - 8) 

(-∞, -7] ∪ [7, ∞) ∪ (8, ∞) 

500

Find the domain AND range of: 

y = (x) / (x+6)

Domain: (-∞, -6) ∪  (-6, ∞)

Range: (-∞, 1) ∪ (1, ∞)

Derivatives

100

Find the derivative of:

-1(x+6)^-2

(-1) / (x+6)²

200

Find the derivative of: 

(x² -2)(3x³+7) 

15x⁴ - 18x² + 14x 

300

Find the equation of the normal line AND tangent line for: 

f(x) = 2x² - 9 at (3,9) 

Normal line: y-9 = 12 (x-3) 

Tangent line: y-9 = -1/12 (x-3) 

400

Find the derivative of: 

y= cos³(4x⁵) 

-60x⁴cos²(4x⁵)sin4x⁵

500

Find the derivative of: 

g(t) = t² 2^t 

g'(t) = t² (2^t  ln2) + 2t (2^t)

Limits

100

Find: 

limit of (x² - 4x + 4) 

as x → -2 

16 

200

Find: 

limit of (x² + 12x + 27) / (x² - 9) 

as x → -3 

-1 

300

Find: 

limit of (√x - √5 ) / (x-5) 

as x → 5 

(1) / (2√5) 

400

Find 

limit of (-x² - x +7) / (2x⁴ - 3x + 2) 

as x →  ∞ 

0

500

f(x) = (x² -1) / (x-1)

(A) Where is the function discontinued? 

(B) What is the nature of the discontinuity? (hole, gap, V.A., H.A.) 

(A) The function is discontinued at x=1 

(B) Hole 

Implicit Differentiation

100

Find the derivative using implicit differentiation: 

x⁴ + 2y² = 8 

dy/dx = (-x)³ / y 

200

Find the derivative using implicit differentiation: 

2xy + y² = 6 

dy/dx = (-y) / (x+y) 

300

Find the second derivative of x² + y² = 25 

-25/y³

400

Find the second derivative of: 

4y² + 2 = 3x²

-6/ (16y³)

500

Find the second derivative of: 

5 = 4x² + 5y²

(-4) / (5y³)

Miscellaneous

100

Name all the trig derivatives: 

sin'(x) = _______

cos'(x) = _______

tan' (x) = ________

csc'(x) = ________

sec'(x) = _______

cot'(x) = ________

sin'(x) = cosx

cos'(x) = -sinx

tan' (x) = sec²x

csc'(x) = (-cscx)(cotx)

sec'(x) = secxtanx

cot'(x) = -csc²x

200

Fill in the following derivative rules: 

d/dx (e^u) = ________

d/dx (a^u) = ________

d/dx (ln * u) = ________

d/dx (log_au) = ________

d/dx (e^u) = e^u * u' 

d/dx (a^u) = a^ulna * u'

d/dx (ln * u) = 1/u * u'

d/dx (log_au) = 1/(ulna) * u' 

300

Find: 

limit of (sinx) / (7x) 

as  x → 0 

1/7

400

Find the derivative of: 

f(x) = cos³( x/(x+4) )

(-12cos²xsinx) / (x+4)⁴

500

Discuss the continuity and differentiability of the function: 

f(x) = x^1/3

continuous but NOT differentiable at x = 0