QUESTION TWO: A- (1) the average rate of change of the function y = f(x), when x changes to x + Ax is written…

Question

QUESTION TWO: A- (1) the average rate of change of the function y = f(x), when x changes to x + Ax is written as: B- (1) If y = ax², where a is a real number, then dy/dx =. (2) Find the first derivative of the following functions: i- y = 3x²/(x+1)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the problems from the image:

Top Left (4):

$\lim_{x \to 0} \frac{\sin 3x}{x}$

To evaluate this limit, we use the standard limit $\lim_{u \to 0} \frac{\sin u}{u} = 1$ .
We need to make the argument of the sine function match the denominator. $\lim_{x \to 0} \frac{\sin 3x}{x} = \lim_{x \to 0} \frac{3 \sin 3x}{3x}$
Let $u = 3x$ .
As $x \to 0$ , $u \to 0$ . $3 \lim_{u \to 0} \frac{\sin u}{u} = 3 \times 1$
Calculate the final value.
$$ 3 \times 1 = \boxed{3} $$
Top Right (2):
- Find the first derivative of the function $y = f(x) = x^2$ , from the first principles.
The definition of the derivative from first principles is
$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$
Substitute $f(x) = x^2$ into the formula.
$$ f(x+h) = (x+h)^2 = x^2 + 2xh + h^2 $$
Calculate $f(x+h) - f(x)$ .
$$ f(x+h) - f(x) = (x^2 + 2xh + h^2) - x^2 = 2xh + h^2 $$
Substitute this back into the limit expression.
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

Factor out $h$ from the numerator and simplify.

$$
f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} = \lim_{h \to 0} (2x + h)
$$

Evaluate the limit as $h \to 0$ .
$$ f'(x) = 2x + 0 = \boxed{2x} $$
QUESTION TWO:

A-(1) The average rate of change of the function $y = f(x)$ , when $x$ changes to $x + \Delta x$ is written as:
The average rate of change is defined as the change in $y$ divided by the change in $x$ .
$\frac{\Delta y}{\Delta x} = \boxed{\frac{f(x+\Delta x) - f(x)}{\Delta x}}$
B-(1) If $y = ax^2$ , where $a$ is a real number, then $\frac{dy}{dx} = \_$
Apply the power rule for differentiation, which states that $\frac{d}{dx}(cx^n) = cnx^{n-1}$ .
Here, $c=a$ and $n=2$ .
$\frac{dy}{dx} = a \cdot 2x^{2-1}$
Simplify the expression.
$\frac{dy}{dx} = \boxed{2ax}$
B-(2) Find the first derivative of the following functions:
- i- $y = \frac{3x^2}{x+1}$
Use the quotient rule for differentiation, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$ .
Let $u = 3x^2$, so $u' = \frac{d}{dx}(3x^2) = 6x$. Let $v = x+1$, so $v' = \frac{d}{dx}(x+1) = 1$.
Substitute $u, u', v, v'$ into the quotient rule formula.
$$ \frac{dy}{dx} = \frac{(6x)(x+1) - (3x^2)(1)}{(x+1)^2} $$

Expand the numerator.

$$
\frac{dy}{dx} = \frac{6x^2 + 6x - 3x^2}{(x+1)^2}
$$

Combine like terms in the numerator.

$$
\frac{dy}{dx} = \boxed{\frac{3x^2 + 6x}{(x+1)^2}}
$$
(Alternatively, you can factor the numerator: $\frac{3x(x+2)}{(x+1)^2}$)

QUESTION TWO: A- (1) the average rate of change of the function y = f(x), when x changes to x + Ax is written as: B- (1) If y = ax², where a is a real number, then dy/dx =. (2) Find the first derivative of the following functions: i− y = 3x²/(x+1)

Step-by-step solution

To evaluate this limit, we use the standard limit $\lim_{u \to 0} \frac{\sin u}{u} = 1$ .

Let $u = 3x$ .

Calculate the final value.

The definition of the derivative from first principles is

Substitute $f(x) = x^2$ into the formula.

Calculate $f(x+h) - f(x)$ .

Substitute this back into the limit expression.

Factor out $h$ from the numerator and simplify.

Evaluate the limit as $h \to 0$ .

Apply the power rule for differentiation, which states that $\frac{d}{dx}(cx^n) = cnx^{n-1}$ .

Simplify the expression.

Use the quotient rule for differentiation, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$ .

Substitute $u, u', v, v'$ into the quotient rule formula.

Expand the numerator.

Combine like terms in the numerator.

Still stuck on this one?

More Mathematics Questions

QUESTION TWO: A- (1) the average rate of change of the function y = f(x), when x changes to x + Ax is written as: B- (1) If y = ax², where a is a real number, then dy/dx =. (2) Find the first derivative of the following functions: i− y = 3x²/(x+1)

Step-by-step solution

To evaluate this limit, we use the standard limit lim⁡u→0sin⁡uu=1\lim_{u \to 0} \frac{\sin u}{u} = 1limu→0​usinu​=1.

Let u=3xu = 3xu=3x.

Calculate the final value.

The definition of the derivative from first principles is

Substitute f(x)=x2f(x) = x^2f(x)=x2 into the formula.

Calculate f(x+h)−f(x)f(x+h) - f(x)f(x+h)−f(x).

Substitute this back into the limit expression.

Factor out hhh from the numerator and simplify.

Evaluate the limit as h→0h \to 0h→0.

Apply the power rule for differentiation, which states that ddx(cxn)=cnxn−1\frac{d}{dx}(cx^n) = cnx^{n-1}dxd​(cxn)=cnxn−1.

Simplify the expression.

Use the quotient rule for differentiation, which states that if y=uvy = \frac{u}{v}y=vu​, then dydx=u′v−uv′v2\frac{dy}{dx} = \frac{u'v - uv'}{v^2}dxdy​=v2u′v−uv′​.

Substitute $u, u', v, v'$ into the quotient rule formula.

Expand the numerator.

Combine like terms in the numerator.

Still stuck on this one?

More Mathematics Questions

To evaluate this limit, we use the standard limit $\lim_{u \to 0} \frac{\sin u}{u} = 1$ .

Let $u = 3x$ .

Substitute $f(x) = x^2$ into the formula.

Calculate $f(x+h) - f(x)$ .

Factor out $h$ from the numerator and simplify.

Evaluate the limit as $h \to 0$ .

Apply the power rule for differentiation, which states that $\frac{d}{dx}(cx^n) = cnx^{n-1}$ .

Use the quotient rule for differentiation, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$ .