Mathematics, 14.01.2021 02:50 alexanderickh1

Need help please its Calculus. Ill give the 5 stars as well.

Answers: 1

Show answers

Answers

Answer from: pickelswolf5938

$\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1$

General Formulas and Concepts:

Pre-Algebra

Order of OperationsEquality Properties

Algebra I

FunctionsFunction NotationExponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Algebra II

Natural logarithms ln and Euler's number e

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Slope Fields

Separation of VariablesSolving Differentials

Integrals

Antiderivatives

Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

U-Substitution

Logarithmic Integration: $\displaystyle \int {\frac{1}{u}} \, dx = ln|u| + C$

Step-by-step explanation:

*Note:

When solving differential equations in slope fields, disregard the integration constant C for variable y.

Step 1: Define

$\displaystyle \frac{dy}{dx} = x^2(y - 1)$

$\displaystyle f(0) = 3$

Step 2: Rewrite

Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.

[Separation of Variables] Rewrite Leibniz Notation: $\displaystyle dy = x^2(y - 1) \ dx$ [Separation of Variables] Isolate y's together: $\displaystyle \frac{1}{y - 1} \ dy = x^2 \ dx$

Step 3: Find General Solution Pt. 1

[Differential] Integrate both sides: $\displaystyle \int {\frac{1}{y - 1}} \, dy = \int {x^2} \, dx$ [dx Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle \int {\frac{1}{y - 1}} \, dy = \frac{x^3}{3} + C$

Step 4: Find General Solution Pt. 2

Identify variables for u-substitution for dy.

Set: $\displaystyle u = y - 1$ Differentiate [Basic Power Rule]: $\displaystyle du = dy$

Step 5: Find General Solution Pt. 3

[dy Integral] U-Substitution: $\displaystyle \int {\frac{1}{u}} \, du = \frac{x^3}{3} + C$ [dy Integral] Integrate [Logarithmic Integration]: $\displaystyle ln|u| = \frac{x^3}{3} + C$ [Equality Property] e both sides: $\displaystyle e^\bigg{ln|u|} = e^\bigg{\frac{x^3}{3} + C}$ Simplify: $\displaystyle |u| = Ce^\bigg{\frac{x^3}{3}}$ Rewrite: $\displaystyle u = \pm Ce^\bigg{\frac{x^3}{3}}$ Back-Substitute: $\displaystyle y - 1 = \pm Ce^\bigg{\frac{x^3}{3}}$ [Equality Property] Isolate y: $\displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1$

General Form: $\displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1$

Step 6: Find Particular Solution

Substitute in function values [General Form]: $\displaystyle 3 = \pm Ce^\bigg{\frac{0^3}{3}} + 1$ Simplify: $\displaystyle 3 = \pm C + 1$ [Equality Property] Isolate C: $\displaystyle 2 = \pm C$ Rewrite: $\displaystyle C = 2$ Substitute in C [General Form]: $\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1$

∴ our particular solution is $\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1$ .

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentials and Slope Fields

Book: College Calculus 10e

Answer from: Quest

the answer to the question is a

it has the same slope and the same y-intercept.

:) will give brainliest! < 3 the pics down below : )

Answer from: Quest

step-by-step explanation:

∆abc has the points a(1, 7), b(-2, 2), and c(4, 2) as its vertices. formed with the point d(1, 2) as its third vertex, then ∆abd is triangle.

Another question on Mathematics

Mathematics, 21.06.2019 18:40

That table shows measurements (in invhes$ from cubes with different side lengths. which pairs of variables have a linear relationship? check all that apply

Answers: 3

Answer

Mathematics, 21.06.2019 20:00

Write the point-slope form of the line passing through (2, -12) and parallel to y=3x.

Answers: 3

Answer

Mathematics, 21.06.2019 23:00

12 out of 30 people chose their favourite colour as blue caculate the angle you would use for blue on a pie chart

Answers: 1

Answer

Mathematics, 22.06.2019 00:20

What is the equation of a line that passes through points (0, 4) and (-4,-8)?