## Questions and Answers

What is the purpose of finding the limit of a function as x approaches a certain value?

To determine the value of the function as x approaches a specific value

What is the formula to find the average gradient between two points?

m = (y2 - y1) / (x2 - x1)

What is the definition of a derivative from first principles?

f'(x) = lim(h → 0) [f(x+h) - f(x)]/h

What is the purpose of finding the derivative of a function?

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What is the value of lim(x → 1) f(x) for f(x) = 2x^2 + 4?

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What is the gradient of the line passing through points (1, 2) and (3, 8)?

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What is the derivative of f(x) = x^2 using the definition from first principles?

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What is the concept of finding the limit of a function as x approaches a certain value?

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What is the purpose of the average gradient formula?

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What is the result of substituting x with values closer and closer to a certain value in a function?

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What is the primary purpose of using the limit equation?

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What is the average gradient between two points (x1, y1) and (x2, y2) representing?

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Which of the following is an application of the derivative from first principles?

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What is the effect of substituting x with values closer and closer to a in the function f(x)?

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What is the purpose of the formula m = (y2 - y1) / (x2 - x1)?

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What does the limit equation lim x→a f(x) represent?

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What is the derivative of the function f(x) = 2x^3?

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What is the limit of the expression (x+h)^2 - x^2 / h as h approaches 0?

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What is the derivative of the function f(x) = x^4 using the differentiation rule?

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What is the purpose of the differentiation rule in calculus?

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What is the derivative of the function f(x) = 3x^2?

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What is the formula for differentiating the function f(x) = kx^n?

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What is the derivative of the function f(x) = x^2 + 2x, using the differentiation rule?

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What is the formula to find the derivative of a function f(x) = kx^n?

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What does the concept of a limit of a function as x approaches a certain value represent?

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What is the formula to find the average gradient between two points (x1, y1) and (x2, y2)?

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What is the derivative of the function f(x) = x^2 using the definition from first principles?

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What is the purpose of the concept of a limit in calculus?

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## Study Notes

### Calculus Essentials

#### The Concept of a Limit

- The limit equation is:
`lim x→a f(x)`

- To find the limit of
`f(x)`

as`x`

approaches a value`a`

, substitute`x`

with values closer and closer to`a`

from both the left and the right, observing if`f(x)`

approaches a specific value. - Example: For
`f(x) = 2x^2 + 4`

,`lim x→1 f(x) = 2(1)^2 + 4 = 6`

#### Average Gradient Between Two Points

- The average gradient equation is:
`m = (y2 - y1) / (x2 - x1)`

- To find the gradient (slope) between two points
`(x1, y1)`

and`(x2, y2)`

, use the formula to calculate the rise over run. - Example: For points
`(1, 2)`

and`(3, 8)`

,`m = (8 - 2) / (3 - 1) = 6/2 = 3`

#### Derivative from First Principles

- The derivative equation is:
`f'(x) = lim h→0 [f(x+h) - f(x)] / h`

- This definition calculates the derivative of
`f(x)`

at a point`x`

, representing the function's rate of change or the slope of the tangent at that point. - Example: For
`f(x) = x^2`

,`f'(x) = lim h→0 [(x+h)^2 - x^2] / h = lim h→0 (2xh + h^2) / h = lim h→0 (2x + h) = 2x`

#### Differentiation Rules

- The differentiation rule equation is:
`d/dx [kx^n] = nkx^(n-1)`

- To differentiate a function
`kx^n`

, multiply the exponent`n`

by the coefficient`k`

and decrease the exponent by one. - Example: For
`f(x) = 3x^4`

,`f'(x) = 4 × 3x^(4-1) = 12x^3`

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## Description

Learn how to work with limits, gradients, and derivatives in calculus with examples and equations. Understand the concept of a limit and how to apply it to find the limit of a function as x approaches a value. Practice with examples from calculus.