Need help with calculus questions

Question

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Answer 1

3)

Apply derivative with respect of r.

 

Answer 2

(1).

The function is y = 2x³ - 4x² + x.

Derivative with respect to 'x'.

f^,(x) = 6x² - 8x + 1.

TEST FOR INCREASING AND DECREASING FUNCTIONS :

If f^,(x) > 0 (Positive) for all x in (a, b), then f(x) is increasing on [a, b].

If f^,(x) < 0 (Negative) for all x in (a, b), then f(x) is decreasing on [a, b].

To find the critical or key numbers, to make the first derivative equal to zero or f ' (x) = 0.

6x² - 8x + 1 = 0

a = 6, b = -8 and c = 1

x = [-b ± √(b² - 4ac)]/2a

x = [-(-8) ± √{(-8)² - 4(6)(1)}] / 2(6)

x = [8 ± √{64 - 24}] / 12

x = [8 ± 2√{10}] / 12

x = [4 ± √10] / 6

x = (4 + √10) / 6 and x = (4 - √10) / 6

x = 1.1937 and x = 0.1396

The key numbers are x = 0.1396 and x = 1.1937.

Test intervals      x - Value                Polynomial or f^,(x) value            Conclusion

(- ∞, 0.14)              x = 0            6(0)² - 8(0) + 1 = 0 - 0 + 1 = 1 > 0             Increasing

(0.14, 1.19)             x = 1            6(1)² - 8(1) + 1 = 6 - 8 + 1 = -1 < 0            Decreasing

(1.19, ∞)                x = 2            6(2)² - 8(2) + 1 = 24 - 16 + 1 = 9 > 0          Increasing

So, f(x) is increasing on the interval (- ∞, 0.14) and (1.19, ∞) and decreasing on the interval (0.14, 1.19).

 

Answer 3

(2).

The equation : 3y² - y = 2x².

Answer 4

4) First method : find the derivative by using fundamental theorem of derivatives.

• 

Apply the formula

Evaluate the limits.

Second method :

f(x) = sin(x)

f(x) = sin(x) (1)

Apply derivative on each side.

Apply product rule d/dx(uv) = uv' + vu'

f'(x) = sin(x)(0) + 1(cos(x))

f'(x) = cos(x)

Answer 5

5)

First method : find the derivative by using fundamental theorem of derivatives.

 

• 

  
  

  

  

  Apply the formula

  

  

  

  

• Evaluate the limits.

  

  

  

• Second method :

  f(x) = cos(x)

  f(x) = cos(x) (1)

  Apply derivative on each side.

  Apply product rule d/dx(uv) = uv' + vu'

  f '(x) = cos(x)(0) + 1(-sin(x))

  f '(x) = - sin(x)

Answer 6

6)

Apply reciprocal identities.

Apply quotient rule in derivatives

7)

Apply quotient rule in derivatives

8)

Apply pythagorean identity

 

Answer 7

9)

First method : find the derivative by using fundamental theorem of derivatives.

Second method :

Apply derivative on each side.

Apply product rule d/dx(uv) = uv' + vu'

.

Answer 8

10)

First method : find the derivative by using fundamental theorem of derivatives.

Let

Second method :

Apply derivative on each side.

Apply product rule d/dx(uv) = uv' + vu'

.