Calculus Answers

Find the centroid of the region bounded by y=x² and y=2

The function f, defined by f(x)= |x-2| is differentiable in [0,1].

True or false with full explanation

Use the method of implicit differentiation to determine the derivative of the following functions:

1. xsiny +ysinx=1

2. y + xcosy=x^2 y

which of the expressions below are equal to

"3\u20224\u20228 - 5\u20226\u202216 + 7\u20228\u202232 - ..."

when summed to the "k" terms?

1)

"k""+ 1"

"\u03a3" "a(a+1)(-1)" ^a+1 "2^a"

"a =" "3"

2)

"k"

"\u03a3" "(2a + 1) (2a + 2) 2" ^{a + 2}

"a =" "1"

3)

"k""+ 1"

"\u03a3" "4(-1)^a a(2a-1)2^a"

"a =" "2"

4)

"k"

"\u03a3" "(-1)^a a(a+1)(a+5)"

"a =" "3"

5) none of the above

which of the expressions below are equal to

"4\u20221 + 7\u20224 + 10\u20229 +13\u202216..."

when summed to the "k" terms?

1)

"k"

"\u03a3" "(a^3 + 3a^2)"

"a = 1"

2)

"k"

"\u03a3" "(3a-1)(a^2)"

"a = 1"

3)

"k"

"\u03a3" "(3a^3 + a^2)"

"a = 1"

4)

"\ufeffk(k+1)(9k^2 + 13k +2)"

_________________________

"12"

5)

"k(k+1)(9k^2+2k+13)"

___________________________

4

Given series:

"n"

"\ufeffSn = \u03a3 \ufeff" "\ufeff((k+1)^2 - k^2)"

"\ufeffk=1"

which of the following statements are true?

1. it is a telescoping series
2. it sums to n(n+1)
3. it sums to n(n+2)
4. it sums to n(n+1) - n
5. it sums to the same value as:

"n"

"\u03a3" "(2k + 1)"

 "k = 1"

6) it sums to the same value as:

"n + 1"

"\u03a3" "(2k)"

"k = 2"

Consider the series:

"\u221e"

"\u03a3" [ "(x^2 + 9)" / 25 "]^a"

"a = 1"

The values of "y" of which this series converges form an internal "(-k, k)" .

Give the value for "k"

Consider a rectangle with perimeter 28 (units). Let the width of the 

rectangle be w (units) and let the Area of the region enclossed by the 

rectangle be A ( square units). 

 Express A as a function of w and state the domain and range of the 

 function.

1. The energy 𝑖, of an inductor with inductance 𝐿 is given by

𝑖= 1/2𝐿∫¹₀ 𝑡^2𝑒^−𝑡 𝑑𝑡

For 𝐿=(1 ×10−3)𝐻, Find 𝑖.

3. The distance travelled by a train on a straight track in the first two seconds is given by

𝑠= ∫²₀ 20(1−𝑒^−𝑡)𝑑𝑡20

Find the distance travelled in Metres