determine the slope of the tangent for the following function. Then determine the equation of the tangent line at x=1
F(x) = x
F(x) = x^4
For F(x) = x, the first derivative - which is the slope of the line tangent to a curve - would be:
F'(x) = 1
For the second part, it doesn't matter what value x is, F'(x) is always 1.
For F(x) = x^4, the first derivative - which is the slope of the line tangent to a curve - would be:
F'(x) = 3x^3
For the second part, you just plug in 1 for x:
F'(x) = 3(1)^3 = 3
HOLD ON!!! MY BAD!
The second function's derivative is wrong. Brain fart on my part.
For F(x) = x^4, the first derivative - which is the slope of the line tangent to a curve - would be:
F'(x) = 4x^3
For the second part of the question, you just plug in 1 for x:
F'(x) = 4(1)^3 = 4
To determine the slope of the tangent to a function at a specific point, we can use calculus. The derivative of the function gives us the slope of the tangent line at any given point.
1. For the function f(x) = x, we take the derivative to find the slope:
f'(x) = 1, because the derivative of x with respect to x is 1.
Therefore, the slope of the tangent line to f(x) = x is 1.
2. For the function f(x) = x^4, we also take the derivative to find the slope:
f'(x) = 4x^3, because the derivative of x^4 with respect to x is 4x^3.
Therefore, the slope of the tangent line to f(x) = x^4 is 4x^3.
To determine the equation of the tangent line at x = 1, we need the slope and a point on the line.
1. For f(x) = x, we already know the slope is 1. To find the y-coordinate of the point on the line, substitute x = 1 into the function:
f(1) = 1.
Therefore, the point on the tangent line at x = 1 is (1, 1).
2. For f(x) = x^4, we know the slope is 4x^3. To find the y-coordinate of the point on the line, substitute x = 1 into the function:
f(1) = 1^4 = 1.
Therefore, the point on the tangent line at x = 1 is (1, 1).
Now that we have the slope (1) and a point (1, 1) on the tangent line for both functions, we can write the equations of the tangent lines using the point-slope form:
1. For f(x) = x:
y - y1 = m(x - x1), where (x1, y1) is the point (1, 1):
y - 1 = 1(x - 1)
y - 1 = x - 1
y = x
2. For f(x) = x^4:
y - y1 = m(x - x1), where (x1, y1) is the point (1, 1):
y - 1 = 4(1^3)(x - 1)
y - 1 = 4(x - 1)
y = 4x - 3
Therefore, the equations of the tangent lines at x = 1 for the functions f(x) = x and f(x) = x^4 are y = x and y = 4x - 3, respectively.