The slope of the the tangent line to a curve at a point. Calc derivative help

Mathematics

1 answer:

pickupchik [31]3 years ago

5 0

Answer:

y = 108x + 216

Step-by-step explanation:

given

y = 4x³ then $\frac{dy}{dx}$ = 12x² = $m_{tgt}$

x = 3 ⇒ m = 12(- 3)² = 108

equation of tangent in slope- intercept form is

y = mx + b ( m is the slope and b the y- intercept ), hence

y = 108x + b

To find b substitute (- 3, - 108) into the equation

- 108 = -324 + b ⇒ b = - 108 + 324 = 216

y = 108x + 216 ← equation of tangent

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Answer:

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from point a leah walked 40 yards south, 60 yards west, 10 yards north, and 20 yards east to point b. what is the length, in yar

nataly862011 [7]

Let's say $A = (0, 0)$ . Let's find point $B$ so that we can find $\overline{AB}$ .

Leah walks 40 yards south. $B = (0, 0 - 40) = (0, -40)$
Leah walks 60 yards west. $B = (0 - 60, -40) = (-60, -40)$
Leah walks 10 yards north. $B = (-60, -40 + 10) = (-60, -30)$
Leah walks 20 yards east. $B = (-60 + 20, -30) = (-40, -30)$

We have found that $B = (-40, -30)$ .

Now think about this scenario visually. We started at the center of something, which we call point $A$ , and then started moving around until we got to point $B$ . We can then form line $\overline{AB}$ between the points. However, realize that we can actually make a triangle. Just think of one of the legs as part of the x-axis and the other leg as part of the y-axis.

We can find the length of these parts, which is simply the absolute value of the coordinates of point $B$ . It may be a little hard to think about, but essentially, we can form a triangle with sides that consist of part of the x-axis, part of the y-axis, and $\overline{AB}$ . We also know that the lengths of the legs are 40 and 30.

Since we are given the two lengths of the legs on the triangle and trying to find the length of the hypotenuse, we can use the Pythagorean Theorem. This states:

$a^2 + b^2 = c^2$