find an equation of the line tangent to the equation at the given point.
f(x)=5x^2 at x=10
To find the equation of the line tangent to the function f(x) = 5x^2 at x = 10, you need to find the value of the slope of the tangent line at x = 10.
The slope of a tangent line can be found using the derivative of the function. Taking the derivative of f(x) = 5x^2:
f'(x) = d/dx (5x^2)
= 10x
Now, substitute x = 10 into the derivative to find the slope of the tangent line:
f'(10) = 10 * 10
= 100
So, the slope of the tangent line is 100.
To find the equation of the tangent line, you need a point on the line. The point of tangency is (10, f(10)). Calculating f(10):
f(10) = 5 * (10^2)
= 5 * 100
= 500
Therefore, the point of tangency is (10, 500).
Now, we can use the point-slope form of a line to find the equation of the tangent line:
y - y1 = m(x - x1)
where (x1, y1) is the point of tangency and m is the slope of the tangent line.
Substituting the values:
y - 500 = 100(x - 10)
Simplifying and rearranging this equation gives:
y - 500 = 100x - 1000
Finally, rearrange the equation to the standard form:
100x - y = 500 - 1000
100x - y = -500
Hence, the equation of the line tangent to f(x) = 5x^2 at x = 10 is 100x - y = -500.
To find the equation of the line tangent to the equation f(x) = 5x^2 at the given point x = 10, you can follow these steps:
Step 1: Find the derivative of f(x)
The derivative of f(x) represents the slope of the tangent line to the function at any given point. In this case, the derivative of f(x) = 5x^2 can be found using the power rule for differentiation:
f'(x) = 10x
Step 2: Find the slope at x = 10
To find the slope of the tangent line at x = 10, substitute the value of x into the derivative equation:
f'(10) = 10(10) = 100
So, the slope of the tangent line at x = 10 is 100.
Step 3: Find the equation of the tangent line
Since we know the slope of the tangent line and a point on the line (10, f(10)), we can use the point-slope equation form:
y - y₁ = m(x - x₁)
where (x₁, y₁) represents the given point on the line, and m represents the slope.
Substituting the values into the equation, we have:
y - f(10) = 100(x - 10)
Now, we need to evaluate f(10):
f(10) = 5(10)^2 = 500
So, the equation becomes:
y - 500 = 100(x - 10)
Simplifying, we get:
y - 500 = 100x - 1000
Rearranging the equation, we obtain the final result:
y = 100x - 500
when x=10 , f(10)=500
so your point is (10,500)
f'(x) = 10x
then when x=10 , slop = 100
Now you have a slope of 100 and a point of (10,500)
Use the method you learned to find the equation of the tangent (a straight line)