pratice final A, TI-89 titatnium calculator may be used if needed. whichever is easier for me. please help

Question

Answer 1

(30)

The radius of the circle is 10 mts.

Rate of change in radius of the circle (dr/dt) is 0.2 mts.

We know that

Area of the circle A is πr²

To calculate the rate of change in area, we differentiate above with respect to time t.

(dA/dt) = π (d(r²)/dt)

(dA/dt) = π(2r * dr/dt)

dA/dt = π * (2*10*0.2)

dA/dt = π (2 * 2)

dA/dt = 4π

Therefore rate of change in area of the circle is dA/dt = 4π m²/sec.

Option (c) is the correct answer.

Answer 2

(31)

The function is f(x) = x^4+2x²

f'(x) = 1

Apply derivative to the function f(x) with respect to x on both sides.

f'(x) = 4x³+2(2x)

f'(x) = 4x³ + 4x

Given that f'(x) = 1

4x³ + 4x = 1

4x³ + 4x - 1 = 0

Roots of the qudratic polynomial are 0.2367, -0.11837+1.0208i and -0.11837 - 1.0208i.

As we do not consider the imaginary values, so x = 0.2367.

Substitute x = 0.2367 in f(x)

f(x) = (0.2367)^4 + 2(0.2367)²

f(x) = 0.1151

So the point is (0.2367, 0.1151)

The tangent line equation is in the form of y = mx + c

f'(x)  = 1 and (0.2367, 0.1151) passes through the line.

y = x + c

0.1151 = 0.2367 + c

c = -0.1216

c = -0.122

then line equation is y= x - 0.122

Therefore the tangent line equation is y= x - 0.122.

Option (d) is the correct answer.

Answer 3

(32)

The cost price function is c(x) = 4x²+4x-80

Where x is the number of computers sold.

The Selling price of each computer is $1012

Total selling price = (Selling price of each computer )* (Number of computers sold ) = 1012x

Profit = selling price - cost price

Profit = 1012x - (4x²+4x-80)

P(x) = -4x² + 1008x + 80

Apply derivative with respect to x.

P'(x) = -8x + 1008

To find the maximum profit, P'(x) = 0

-8x + 1008 = 0

8x = 1008

x = 126

Therefore number of computers must be sold to obtain maximum profit is 126 units.

Option (b) is the correct answer.

Answer 4

(29)

The system of equation are

x + z = 3

x+y-2z = 10

y-3z = 8

Let us write the equation in the form of matrix AX = B

Let us write the augmented matrix

Now apply

Apply

Here the rank of Augmented matrix [A|B] is 3

Rank of matrix A is 2

Rank of [A] < Rank of [A|B] then AX = B is inconsistent and No solutions exist.

Therefore the system has no solutions

Option (e) is the correct answer.