People firstly believe that the planets move in a circular orbit until Newton came up with his hypothesis by inventing calculus so that we could understood and calculated planetary orbits and their accuracy.
Explanation:
Everyone assumed the planets were perfect circles until Newton came up with an idea. Slowly people would make maps of the orbits that added circles on circles, and they could never really explain about the movement of the planet. They simply say that planets move on circles but they lacked the math to explain or prove it. Then Newton came up with an idea of inventing calculus so that we could understood and calculated planetary orbits and their accuracy.
Firstly people used their observations and say that the orbits looked like circles, then they developed their models and did the math, and proposed their hypothesizes which were wrong, until Newton came along and tried to match a model that used elliptical orbits and invented the math that allowed him to make predictions with it. His model worked for most planets.
However he could not explain about the planet Mercury for instance since it was a very strange orbit. Then after the Einstein's theory of General Relativity he could also explain very deeply about it.
Scientists and Astronomers made hypothesizes that there was another planet orbiting too close to the sun to see with telescopes, called Vulcan, that explained mercury's orbit before Einstein's theory. Then long after we had telescopes which was good enough to see if there was a planet orbiting closer to the sun than mercury.
Answer:
(a)
(b)
(c) Simplifying first
General Formulas and Concepts:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]:
Derivative Rule [Quotient Rule]:
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify.</em>
<em />
<u>Step 2: Differentiate Way 1</u>
- Derivative Rule [Quotient Rule]:
- Rewrite [Derivative Property - Addition/Subtraction]:
- Derivative Rule [Product Rule]:
- Rewrite [Derivative Rule - Multiplied Constant]:
- Derivative Rule [Basic Power Rule]:
- Simplify:
- Simplify:
- Simplify:
- Simplify:
- Rewrite:
∴ we find the derivative of the given function but it is a tedious method of computation.
<u>Step 3: Differentiate Way 2</u>
- [Function] Rewrite:
- [Function] Simplify:
- [Derivative] Rewrite [Derivative Property - Addition/Subtraction]:
- Rewrite [Derivative Property - Multiplied Constant]:
- Derivative Rule [Basic Power Rule]:
∴ we find the derivative of the given function <em>and </em>it is less complex and faster. We can conclude that simplifying first appears to be simpler for this problem.
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
We have been given that in ΔBCD, the measure of ∠D=90°, the measure of ∠C=42°, and CD = 7.5 feet. We are asked to find the length of DB to nearest tenth of foot.
First of all, we will draw a right triangle using our given information.
We can see from the attachment that DB is opposite side to angle C and CD is adjacent side to angle.
We know that tangent relates opposite side of right triangle to adjacent side of right triangle.
Therefore, the length of DB is approximately 6.8 feet.
The answer is A.9
do u need how i got the answer