Calc Final

Limits

1

What is the limit as x approaches to infinity for sin(x)?

• DNE

2

When does the limit not exist?

• oscillating behavior

• Unbounded behavior

• Removable discontinuity

The first and second option

3

What is the limit as x approaches 3 for (x^2 -3x +2)/(x-2)?

2

4

f(x)={5x-1 x<1, 10-2x x≥ 1}

Does f'(1) exist and is MVT satisfied on (-2,2)

f’(1) = -2 and no, discontinuity -> theorem not satisfied 

5

There are two graphs:

f(x) is going from the bottom to up with a limit of -3 at x=2. From up to bottom, it has a limit of 4 at x=2.

g(x) is approaching from the bottom on both sides to 2 when x=2.

What is the limit of f(g(x)) at x approaches 2?

1. Limit of f(g(2)) is -3

Derivatives

1

g(x)=(6x²)/(2-x)

Solution: g'(x)=(24x-6x²)/(2-x)²

2

Differentiate h(t)=(t³)-(t²)sin(t)

h'(t)=(3t²)-2tsin(t)-(t²)(cos(t))

3

Z(v)= (v+tanv)/(1+cscv)

Z'(v)= ((1+sec²(v))(1+cscv)-(v+tanv)(-csc(v)cot(v))/(1+cscv)²

4

Given that x=-2, y=1, and x'=-4, determin y' for the following equation.

6y²+x²=2-x³e^4-4y

y'=8/11

5

g(x)=(ln(x²+1)-tan^-1(6x))¹⁰

g'(x)=10(ln(x²+1)-tan^-1(6x)^9 ((2x/(x²+1)-(6/(36x²+1)

Series

1

1. What is the coefficient of 𝑥²  in the Taylor series for 𝑠𝑖𝑛(𝑥)² about 0 ?

1

2

2. Determine if the series converges or diverges:

The summation of (-1)^n-1 /(7+2n). (n=1 and ontop of sumation is infinity)

So, both of the conditions in the Alternating Series Test are met and so the series is convergent. (limit is 0 and it is decreasing)

3

2. The coefficient of (𝑥 − (𝜋 /4) )^3 in the Taylor series about 𝜋/4 of 𝑓(𝑥) = 𝑐𝑜𝑠𝑥 is

1/(6√2)

4

4. Let 𝑓 be the function given by 𝑓(𝑥) = cos(2𝑥). Write the first four nonzero terms and the general term of the Taylor series for 𝑓 about 𝑥 = 0.

𝟏 − ((𝟐𝒙)²)/ 𝟐! + ((𝟐𝒙)⁴)/𝟒! − ((𝟐𝒙)⁶)/𝟔! + ⋯ + (−𝟏)ⁿ ((𝟐𝒙)²ⁿ)/(𝟐𝒏)! 

5

5. At t≥0, a particle moving in the xy plane has the velocity vector given by v(t)= <4e^-t, sin(1+√(t)). What is the total distance the particle moves between t=1 and t=3?

1.861

Polar

1

1. The polar curve C has equation

2. r = − 2 (cos(θ)sin(θ) ), 0  ≤ θ< 2π .

Find a Cartesian equation for C. 

Answer: (x-1)²+(y+1)²=2

2

1. The curve C has polar equation: r = 6cos(3θ) ,   − π< θ≤ π .

Find the exact value of area enclosed by the C , for (-π/6) < θ ≤ (π/6)

Answer: Area: 3π

3

1. The curve C with polar equation

   1. r = √(6) cos (2θ) , 0≤ θ ≤ π/4.

The straight line l is parallel to the initial line and is a tangent to C .

Find an equation of l , giving the answer in the form r =f (θ )

Answer: r= 2/3cosecθ

4

1. The following polar equations are given:

   1. r1=cosθ, 0 ≤ θ ≤ π

   2. r2= 1/(cosθ - sin θ), (-¼)π≤ θ ≤(5/4)π

Find, in exact simplified form, the area of the smaller of the two finite regions, bounded by r1 and r2.

Answer: (3π+ 2)/16

5

1. r²= 2cos2θ, 0≤θ< π/4

   1. The finite region R is bounded by C , L and the half line with equation θ = π/2 (y-axis).

Find the Area of region R.

Answer: (⅛)(3√(3) -4)

Integrals

1

Evaluate ∫ x^(3/2) + 2x + 3 dx

(2/5)x^(5/2) + x² +3x + c

2

Evaluate ∫ x⁷ - 48x¹¹ - 5x¹⁶ dx

(1/8)x⁸ - 4x¹² - (5/17)x¹⁷ + c

3

Evaluate ∫ x³ - ((e^-x - 4)/e^-x) dx

(1/4)x⁴ - x + 4e^x + c

4

Evaluate ∫₄⁹ (√x + (1/3√x) dx

40/3 = 13.333

5

Determine h(t) given that h''(t) = 24t² - 48t + 2, h(1) = -9 and h(-2) = -4

h(t) = 2t⁴ - 8t³ + t² + (100/3)t - (112/3)