find the slopes at the indicated points
y = 3 + 5x - 3x^3 (0,3), (1/2, 41/8), (2,-11)
What about this do you not understand. Take the derivative, evaluate at the designated points.
The slope is
dy/dx = 5 -9 x^2
Plug in the appropriate x values for the slope at each one.
To find the slopes at the indicated points, we need to calculate the derivative of the function y = 3 + 5x - 3x^3 and evaluate it at those points. The derivative gives us the slope of the function at any given point.
Step 1: Find the derivative of the function y = 3 + 5x - 3x^3.
To do this, we differentiate each term of the function with respect to x. The derivative of a constant is 0, the derivative of x to the power of n (x^n) is nx^(n-1), and the derivative of a sum is the sum of the derivatives. Applying these rules, we get:
dy/dx = d(3)/dx + d(5x)/dx - d(3x^3)/dx
Since the derivative of a constant is 0 and d(5x)/dx = 5, we have:
dy/dx = 0 + 5 - d(3x^3)/dx
To differentiate 3x^3, we use the power rule, which states that the derivative of x^n is nx^(n-1). Applying the power rule, we have:
dy/dx = 5 - 3 * d(x^3)/dx
Using the power rule again, we differentiate x^3 to get:
dy/dx = 5 - 3 * 3x^2
Simplifying further, we have:
dy/dx = 5 - 9x^2
Step 2: Evaluate the derivative at the given points.
Now that we have the derivative dy/dx = 5 - 9x^2, we can substitute the x-values from each point into the derivative to find the slopes.
(a) Point (0,3):
Substituting x = 0 into the derivative, we have:
dy/dx = 5 - 9(0)^2
dy/dx = 5 - 9(0)
dy/dx = 5 - 0
dy/dx = 5
The slope at (0,3) is 5.
(b) Point (1/2, 41/8):
Substituting x = 1/2 into the derivative, we have:
dy/dx = 5 - 9(1/2)^2
dy/dx = 5 - 9(1/4)
dy/dx = 5 - 9/4
To simplify this, we need to find a common denominator:
dy/dx = 5 - 9/4
dy/dx = 20/4 - 9/4
dy/dx = (20 - 9)/4
dy/dx = 11/4
The slope at (1/2, 41/8) is 11/4.
(c) Point (2, -11):
Substituting x = 2 into the derivative, we have:
dy/dx = 5 - 9(2)^2
dy/dx = 5 - 9(4)
dy/dx = 5 - 36
dy/dx = -31
The slope at (2, -11) is -31.
To summarize, the slopes at the indicated points are:
(0,3): 5
(1/2, 41/8): 11/4
(2, -11): -31