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Irina18 [472]
3 years ago
14

PLEASE HELP 50 points

Mathematics
2 answers:
inysia [295]3 years ago
7 0

Answer:

  1. Scalene triangle [any side and angle are not equal]
  2. isosceles triangle [two side and two angle are equal]
  3. equilateral triangle [all sides and angle are equal]
  4. right angled isosceles triangle [having one angle 90° and two other side and two angle are equal]
  5. scalene triangle
  6. equilateral triangle
  7. right angled triangle
  8. scalene triangle
  9. right angled isosceles triangle
  10. right angled isosceles triangle
  11. scalene triangle
  12. isosceles triangle
  13. equilateral triangle
  14. right angled triangle
  15. right angled isosceles triangle
  16. isosceles triangle
  17. scalene triangle
  18. equilateral triangle
  19. right angled isosceles triangle
  20. isosceles triangle
  21. equilateral triangle
  22. scalene triangle
  23. isosceles triangle
  24. right angled isosceles triangle
  25. right angled triangle
  26. right angled isosceles triangle
  27. isosceles triangle

Lemur [1.5K]3 years ago
5 0

___________________________________

\huge\underline {{\red{ \boxed{\tt{Answer}}}}}

1.)<u>Scalene ∆</u>

2.)<u>Isosceles ∆</u>

3.)<u>Equilateral ∆</u>

4.)<u>Right angled isosceles ∆</u>

5.)<u>Scalene ∆</u>

6.)<u>Equilateral ∆</u>

7.)<u>Right angled ∆</u>

8.)<u>Scalene ∆</u>

9.)<u>Right angled isosceles ∆</u>

10.)<u>Right angled isosceles ∆</u>

11.)<u>Scalene ∆</u>

12.)<u>Isosceles ∆</u>

13.)<u>Equilateral ∆</u>

14.)<u>Right angled ∆</u>

15.)<u>Right angled isosceles ∆</u>

16.)<u>Isosceles ∆</u>

17.)<u>Scalene ∆</u>

18.)<u>Equilateral ∆</u>

19.)<u>Right angled isosceles ∆</u>

20.)Isosceles ∆

21.)<u>Equilateral ∆</u>

22.)<u>Scalene ∆</u>

23.)<u>Isosceles ∆</u>

24.)<u>Right angled isosceles ∆</u>

25.)<u>Right angled ∆</u>

26.)<u>Right angled isosceles ∆</u>

27.)<u>Isosceles ∆</u>

___________________________________

#CarryOnLearning

✍︎ C.Rose❀

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Answer:

a)  \displaystyle \frac{dy}{dx} \bigg| \limits_{x = 0} = -1

b)  \displaystyle \frac{dy}{dx} \bigg| \limits_{x = \frac{\pi}{2}} = -1

General Formulas and Concepts:

<u>Pre-Calculus</u>

  • Unit Circle

<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           \displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)

Derivative Property [Addition/Subtraction]:                                                         \displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]  

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹  

Derivative Rule [Product Rule]:                                                                             \displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)

Derivative Rule [Quotient Rule]:                                                                           \displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}

Derivative Rule [Chain Rule]:                                                                                 \displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

Trigonometric Differentiation

Logarithmic Differentiation

Step-by-step explanation:

a)

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle y = ln \bigg( \frac{1 - x}{\sqrt{1 + x^2}} \bigg)

<u>Step 2: Differentiate</u>

  1. Logarithmic Differentiation [Chain Rule]:                                                     \displaystyle \frac{dy}{dx} = \frac{1}{\frac{1 - x}{\sqrt{1 + x^2}}} \cdot \frac{d}{dx}[\frac{1 - x}{\sqrt{1 + x^2}}]
  2. Simplify:                                                                                                         \displaystyle \frac{dy}{dx} = \frac{-\sqrt{x^2 + 1}}{x - 1} \cdot \frac{d}{dx}[\frac{1 - x}{\sqrt{1 + x^2}}]
  3. Quotient Rule:                                                                                               \displaystyle \frac{dy}{dx} = \frac{-\sqrt{x^2 + 1}}{x - 1} \cdot \frac{(1 - x)'\sqrt{1 + x^2} - (1 - x)(\sqrt{1 + x^2})'}{(\sqrt{1 + x^2})^2}
  4. Basic Power Rule [Chain Rule]:                                                                     \displaystyle \frac{dy}{dx} = \frac{-\sqrt{x^2 + 1}}{x - 1} \cdot \frac{-\sqrt{1 + x^2} - (1 - x)(\frac{x}{\sqrt{x^2 + 1}})}{(\sqrt{1 + x^2})^2}
  5. Simplify:                                                                                                         \displaystyle \frac{dy}{dx} = \frac{-\sqrt{x^2 + 1}}{x - 1} \cdot \bigg( \frac{x(x - 1)}{(x^2 + 1)^\bigg{\frac{3}{2}}} - \frac{1}{\sqrt{x^2 + 1}} \bigg)
  6. Simplify:                                                                                                         \displaystyle \frac{dy}{dx} = \frac{x + 1}{(x - 1)(x^2 + 1)}

<u>Step 3: Find</u>

  1. Substitute in <em>x</em> = 0 [Derivative]:                                                                     \displaystyle \frac{dy}{dx} \bigg| \limit_{x = 0} = \frac{0 + 1}{(0 - 1)(0^2 + 1)}
  2. Evaluate:                                                                                                         \displaystyle \frac{dy}{dx} \bigg| \limits_{x = 0} = -1

b)

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle y = ln \bigg( \frac{1 + sinx}{1 - cosx} \bigg)

<u>Step 2: Differentiate</u>

  1. Logarithmic Differentiation [Chain Rule]:                                                     \displaystyle \frac{dy}{dx} = \frac{1}{\frac{1 + sinx}{1 - cosx}} \cdot \frac{d}{dx}[\frac{1 + sinx}{1 - cosx}]
  2. Simplify:                                                                                                         \displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{d}{dx}[\frac{1 + sinx}{1 - cosx}]
  3. Quotient Rule:                                                                                               \displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{(1 + sinx)'(1 - cosx) - (1 + sinx)(1 - cosx)'}{(1 - cosx)^2}
  4. Trigonometric Differentiation:                                                                       \displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{cos(x)(1 - cosx) - sin(x)(1 + sinx)}{(1 - cosx)^2}
  5. Simplify:                                                                                                         \displaystyle \frac{dy}{dx} = \frac{-[cos(x) - sin(x) - 1]}{[sin(x) + 1][cos(x) - 1]}

<u>Step 3: Find</u>

  1. Substitute in <em>x</em> = π/2 [Derivative]:                                                                 \displaystyle \frac{dy}{dx} \bigg| \limit_{x = \frac{\pi}{2}} = \frac{-[cos(\frac{\pi}{2}) - sin(\frac{\pi}{2}) - 1]}{[sin(\frac{\pi}{2}) + 1][cos(\frac{\pi}{2}) - 1]}
  2. Evaluate [Unit Circle]:                                                                                   \displaystyle \frac{dy}{dx} \bigg| \limit_{x = \frac{\pi}{2}} = -1

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

Unit: Differentiation

Book: College Calculus 10e

4 0
3 years ago
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