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# Mathematics

## Discover

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### Finding Derivative of a Point?

g(x) = 4x² - 2x + 15 ← this is a function

g(5) = 100 - 10 + 15 = 105 → the representative curve of the function passes through (5 ; 105)

g'(x) = 8x - 2 ← this is its derivative

g'(5) = 40 - 2 = 38 ← this is in fact the slope of the tangent line to the curve at point (5 ; 105)

g(x) = 4x² - 2x + 15 ← this is the function → to calculate the derivative:

Lim [g(x₀ + h) - g(x₀)] / h

h → 0

Lim [ { 4.(x₀ + h)² - 2.(x₀ + h) + 15 } - { 4x₀² - 2x₀ + 15 } ] / h

h → 0

Lim [ { 4.(x₀² + 2x₀.h + h²) - 2x₀ - 2h + 15 } - 4x₀² + 2x₀ - 15 ] / h

h → 0

Lim [4x₀² + 8x₀.h + 4h² - 2x₀ - 2h + 15 - 4x₀² + 2x₀ - 15] / h

h → 0

Lim [8x₀.h + 4h² - 2h] / h

h → 0

Lim h.[8x₀ + 4h - 2] / h

h → 0

Lim (8x₀ + 4h - 2) = 8x₀ - 2

h → 0

g'(x₀) = 8x - 2

g'(5) = 40 - 2 = 38 ← this is the same result (above)

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### How far is a 5k?

3 ounces short of 1 cup.

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### How do I solve 15^x-8+5=62 And round 3 places past the decimal?

Exponential and log equations

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### Percentage question ?

The price of gas increases by 25%. By what percentage must a customer reduce the consumption so that the earlier bill on the gas doesn't alter?

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### Solve the equation sin(x+3)=sinx, 0=<x=<2pi to 4 decimal places. The product of the two solutions is:?

Need help with this problem pls

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### graph your equations: 2x + 3y = 7 and y=2x + 27 label them properly, including point of intersection?

2x + 3y = 7 eq1 and y=2x + 27 eq2

plug in eq2 to eq1

2x + 3(2x + 27) = 7

2x + 6x + 81 = 7

8x = 7 - 81

8x = -74

solving for y at eq2

y = 2(-37/4) + 27 = 17/2 Answer//

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