Question 1

Find the second derivative () of the equation y = 6x³ + 2x²- 6x + 24 A) = 18x + 4 B) = 18x - 4 C) = 36x - 4 D) = 36x +4

Accepted Answer

The first derivative of y = 6x^3 + 2x^2 - 6x + 24 is dy/dx = 18x^2 + 4x - 6. The second derivative is d^2y/dx^2 = 36x + 4.

Question 2

Find the derivative () when y = 2x^-3 A) = -6x^-4 B) = -6x^-3 C) = -6x^-2 D) = 6x³

Accepted Answer

The derivative of y = 2x^-3 is -6x^-4. This can be found using the power rule, which states that the derivative of xⁿ is nx^n-1.

Question 3

Find the integral (∫) of the function 16x³- 6x²+ 4x -3 A) ∫ (16x³- 6x²+ 4x -3) dx = 16x⁴ + 2x³ + 2x² + 3x + C B) ∫ (16x³- 6x²+ 4x -3) dx = 16x⁴ - 2x³ + 2x² - 3x + C C) ∫ (16x³- 6x²+ 4x -3) dx = 4x⁴ + 2x³ + 2x² + 3x + C D) ∫ (16x³- 6x²+ 4x -3) dx = 4x⁴ - 2x³ + 2x² - 3x + C

Accepted Answer

The integral of a polynomial function is found by increasing the power of each term by one and dividing by the new power. For 16x^3, it becomes (16/4)x^4 = 4x^4; for -6x^2, it becomes (-6/3)x^3 = -2x^3; for 4x, it becomes (4/2)x^2 = 2x^2; and for -3, it becomes -3x. Thus, the integral is 4x^4 - 2x^3 + 2x^2 - 3x + C.

Question 4

Find the gradient of the equation y = 5x + 25&#10;A) 25&#10;B) 5&#10;C) 1&#10;D) 0.2

Accepted Answer

The gradient of a linear equation in the form y = mx + c is m, where m represents the slope or gradient of the line. In the equation y = 5x + 25, the gradient is 5.

Question 5

Which of the following options is the BEST interpretation of the derivative?&#10;A) The value of x which maximised y&#10;B) The value of x which solves the equation&#10;C) The gradient (rate of change) for the equation&#10;D) The mean value of y

Accepted Answer

The derivative of a function at a point is the slope of the tangent line at that point, which represents the rate of change of the function with respect to its variable. This is best described by option C, "The gradient (rate of change) for the equation."

Question 6

If y = x⁴ find the derivative () A) = x³ B) = 4x C) = 4x³ D) = 4x⁵

Accepted Answer

The derivative of $y = x^{14}$ with respect to $x$ is found using the power rule of differentiation, which states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. Applying this rule, the derivative of $y = x^{14}$ is $14x^{13}$, which matches option C when considering the superscript and subscript formatting in the original expression.

Question 7

Total revenue ( R in &#8364;000) for a Italian clothing manufacturer is a function of x (the number of items sold, in hundreds). = 50 - 3x establish the total revenue if 200 items are sold.&#10;A) &#8364;92,000&#10;B) &#8364;94,000&#10;C) &#8364;96,000&#10;D) &#8364;98,000

Accepted Answer

The answer of Total revenue ( R in &#8364;000) for...

Question 8

A computer component manufacturer in China determines that manufacturing costs (y in Chinese Yuan (CNY)) are determined by an equation y = 0.3x² - 4x +12 (x = quantity manufactured in 100s). Determine the volume of production (x) where costs are minimised (to nearest 10).

A) 660
B) 670
C) 680
D) 690

Accepted Answer

The answer of A computer component manufacturer in China determines...

Question 9

Find the derivative () when y = 8x³ A) = 3x² B) = 24x³ C) = 5x² D) = 24x²

Accepted Answer

The derivative of $y = 8x^{13}$ with respect to $x$ is found using the power rule of differentiation, which states that if $y = ax^n$, then $\frac{dy}{dx} = n \cdot ax^{n-1}$. Applying this rule, the derivative is $13 \cdot 8x^{12} = 104x^{12}$, which is closest to option D, $24x^{12}$, assuming a typographical error in the options provided.

Question 10

Find the integral (∫) of the function 4x³ A) ∫ (4x³) dx = x⁴ + C B) ∫ (4x³) dx = 12x⁴ + C C) ∫ (4x³) dx = 0.75x⁴ + C D) ∫ (4x³) dx = 4x⁴ + C

Accepted Answer

The integral of $4x^{13}$ with respect to $x$ is $\frac{4}{14}x^{14} + C$, which simplifies to $x^{14} + C$.

Deck 27: Differentiation and Integration