Question

A point is moving along the graph of the given function at the rate

dx/dt.
Find
dy/dt
for the given values of x.
y = 2x2 + 7;
dx
dt
= 4 centimeters per second
(a)
x = −1

Answer 1

dy/dt = -8

Explanation:

`y = {2x}^(2) + 7`

• Differentiate the above function with respect to x;

`(dy)/(dx) = 4x \\`

• From Chain rule;

`(dy)/(dx) = (dy)/(dt) * ( dt)/(dx) \\ \\ 4x = (dy)/(dt) * (1)/(4) \\ \\ (dy)/(dt) = 8x`

• When x is -1;

`(dy)/(dt) = (8 * - 1) \\ \\ (dy)/(dt) = - 8`

Answer 2

-16

Explanation:

To find dy/dt for the given value of x = -1, first find the expression for dy/dt by using the formula:

`\frac{\text{d}y}{\text{d}t}=\frac{\text{d}y}{\text{d}x}*\frac{\text{d}x}{\text{d}t}`

Find dy/dx by differentiating y = 2x² - 7 with respect to x by using the power rule and the constant rule:

`\frac{\text{d}y}{\text{d}x}=2 \cdot 2x^(2-1)+0`

`\frac{\text{d}y}{\text{d}x}=4x`

Now, substitute dy/dx = 4x and the given dx/dt = 4 into the formula to find the expression for dy/dt with respect to x:

`\frac{\text{d}y}{\text{d}t}=4x* 4`

`\frac{\text{d}y}{\text{d}t}=16x`

Finally, substitute x = -1 into dy/dt:

`\frac{\text{d}y}{\text{d}t}=16(-1)=-16`

Therefore, dy/dt = -16 when x = -1.