Answer:
(d) ![\displaystyle 12x^3 - 15x^2 + 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%2012x%5E3%20-%2015x%5E2%20%2B%202)
General Formulas and Concepts:
<u>Algebra I</u>
- Functions
- Function Notation
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
![\displaystyle y = 3x^4 - 5x^3 + 2x - 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%203x%5E4%20-%205x%5E3%20%2B%202x%20-%201)
<u>Step 2: Differentiate</u>
- Basic Power Rule:
![\displaystyle y' = 4(3x^{4 - 1}) - 3(5x^{3 - 1}) + 2x^{1 - 1} - 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%204%283x%5E%7B4%20-%201%7D%29%20-%203%285x%5E%7B3%20-%201%7D%29%20%2B%202x%5E%7B1%20-%201%7D%20-%200)
- Simplify:
![\displaystyle y' = 4(3x^3) - 3(5x^2) + 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%204%283x%5E3%29%20-%203%285x%5E2%29%20%2B%202)
- Multiply:
![\displaystyle y' = 12x^3 - 15x^2 + 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%2012x%5E3%20-%2015x%5E2%20%2B%202)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
Answer:
you can put anything to replace x that is greater or equal to 84.
For example:
It just has to be greater or equal to 84.
> = greater than
< = less than
≥ = greater than or equal to
≤ = less than or equal to
Hope this helpss! :)
<span>Vector Equation
(Line)</span>(x,y) = (x,y) + t(a,b);tERParametric Formx = x + t(a), y = y + t(b); tERr = (-4,-2) + t((-3,5);tERFind the vector equation of the line passing through A(-4,-2) & parallel to m = (-3,5)<span>Point: (2,5)
Create a direction vector: AB = (-1 - 2, 4 - 5)
= (-3,-1) or (3,1)when -1 (or any scalar multiple) is divided out.
r = (2,5) + t(-3,-1);tER</span>Find the vector equation of the line passing through A(2,5) & B(-1,4)<span>x = 4 - 3t
y = -2 + 5t
;tER</span>Write the parametric equations of the line passing through the line passing through the point A(4,-2) & with a direction vector of m =(-3,5)<span>Create Vector Equation first:
AB = (2,8)
Point: (4,-3)
r = (4,-3) + (2,8); tER
x = 4 + 2t
y = -3 + 8t
;tER</span>Write the parametric equations of the line through A(4,-3) & B(6,5)<span>Make parametric equations:
x = 5 + 4t
y = -2 + 3t ; tER
For x sub in -3
-3 = 5 + 4t
(-8 - 5)/4 = t
-2 = t
For y sub in -8
-8 = -2 + 3t
(-8 + 2)/3 = t
-2 = t
Parameter 't' is consistent so pt(-3,-8) is on the line.</span>Given the equation r = (5,-2) + t(4,3);tER, is (-3,-8) on the line?<span>Make parametric equations:
x = 5 + 4t
y = -2 + 3t ; tER
For x sub in 1
-1 = 5 + 4t
(-1 - 5)/4 = t
-1 = t
For y sub in -7
-7 = -2 + 3t
(-7 + 2)/3 = t
-5/3 = t
Parameter 't' is inconsistent so pt(1,-7) is not on the line.</span>Given the equation r = (5,-2) + t(4,3);tER, is (1,-7) on the line?<span>Use parametric equations when generating points:
x = 5 + 4t
y = -2 + 3t ;tER
X-int:
sub in y = 0
0 = -2 + 3t
solve for t
2/3 = t (this is the parameter that will generate the x-int)
Sub t = 2/3 into x = 5 + 4t
x = 5 + 4(2/3)
x = 5 + (8/3)
x = 15/3 + (8/3)
x = 23/3
The x-int is (23/3, 0)</span>What is the x-int of the line r = (5,-2) + t(4,3); tER?Note: if they define the same line: 1) Are their direction vectors scalar multiples? 2) Check the point of one equation in the other equation (LS = RS if point is subbed in)What are the two requirements for 2 lines to define the same line?