Math / Calculus

QuestionCompute (a) $\Delta y$ and (b) dy for the given values of $x$ and $\mathrm{dx}=\Delta x$ . $y=7-x^{2}, \quad x=-2, \quad \Delta x=.4$ (a) $\Delta y=$ $\square$ (b) $d y=$ 16 $\square$

Studdy Solution

STEP 1

1. $\Delta y$ represents the actual change in $y$ when $x$ changes by $\Delta x$ .
2. $dy$ represents the approximate change in $y$ using the derivative.
3. The function given is $y = 7 - x^2$ .
4. The initial value of $x$ is $-2$ and $\Delta x = 0.4$ .

STEP 2

1. Calculate $\Delta y$ .
2. Calculate $dy$ .

STEP 3

To calculate $\Delta y$ , we need to find the change in $y$ as $x$ changes from $-2$ to $-2 + 0.4 = -1.6$ .
First, calculate $y$ at $x = -2$ :
$y(-2) = 7 - (-2)^2 = 7 - 4 = 3$
Next, calculate $y$ at $x = -1.6$ :
$y(-1.6) = 7 - (-1.6)^2 = 7 - 2.56 = 4.44$
Now, find $\Delta y$ :
$\Delta y = y(-1.6) - y(-2) = 4.44 - 3 = 1.44$

STEP 4

To calculate $dy$ , we need to use the derivative of $y$ with respect to $x$ .
First, find the derivative $\frac{dy}{dx}$ :
$\frac{dy}{dx} = \frac{d}{dx}(7 - x^2) = -2x$
Evaluate the derivative at $x = -2$ :
$\frac{dy}{dx} \bigg|_{x = -2} = -2(-2) = 4$
Now, calculate $dy$ using $dx = \Delta x = 0.4$ :
$dy = \frac{dy}{dx} \bigg|_{x = -2} \cdot dx = 4 \cdot 0.4 = 1.6$
The solutions are: (a) $\Delta y = \boxed{1.44}$ (b) $dy = \boxed{1.6}$

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