Find the slope of a line tangent to the curve y=2x^2+4x at the point P(−3,6) by finding the limit of the slopes of the secant lines PQ where Q has x-values −2.5, −2.9, −2.99, and −2.999. Sketch the curve and tangent line
The secant from (-3,6) to (-2.5,2.5) has slope (2.5-6)/(-2.5+3) = -7
now do the same calculation for the other values.
To find the slope of the line tangent to the curve at the point P(-3,6), we can first find the slope of the secant lines PQ, where Q has x-values -2.5, -2.9, -2.99, and -2.999. The slope of a secant line between two points is the difference in the y-coordinates divided by the difference in the x-coordinates.
Let's calculate the slopes of the secant lines using the given x-values:
For P(-3,6) and Q(-2.5, y):
Slope = (y₂ - y₁) / (x₂ - x₁)
= (f(x₂) - f(x₁)) / (x₂ - x₁)
= (2x₂² + 4x₂ - 2x₁² - 4x₁) / (x₂ - x₁)
= (2(-2.5)² + 4(-2.5) - 2(-3)² - 4(-3)) / (-2.5 - (-3))
= (-10 + (-10) + 18 + 12) / (2.5)
= 10 / 2.5
= 4
Similarly, calculate the slope of the secant line for the other x-values:
For x = -2.5, slope = 4
For x = -2.9, slope ≈ 4.6
For x = -2.99, slope ≈ 4.96
For x = -2.999, slope ≈ 4.996
By observing the values, we can see that as the x-values of Q get closer to -3, the slopes of the secant lines PQ approach a certain value. To find the slope of the tangent line, we need to find the limit of these slopes as x approaches -3.
lim(x→-3) slope = lim(x→-3) {(2x² + 4x - 2(-3)² - 4(-3)) / (x - (-3))}
= lim(x→-3) {(2x² + 4x - 18 - 12) / (x + 3)}
To evaluate this limit, we can substitute x = -3 into the expression:
lim(x→-3) slope = (2(-3)² + 4(-3) - 18 - 12) / (-3 + 3)
= (18 - 12 - 18 - 12) / 0
= -24 / 0
The limit is indeterminate form. To find the slope of the tangent line, we can differentiate the given equation y = 2x² + 4x to find the derivative.
Differentiating y = 2x² + 4x with respect to x:
dy/dx = 4x + 4
Now we substitute x = -3 into the derivative to find the slope of the tangent line at (-3, 6):
slope = 4(-3) + 4
= -12 + 4
= -8
Therefore, the slope of the line tangent to the curve y = 2x² + 4x at the point P(-3, 6) is -8.
To sketch the curve and tangent line, we can plot the points P(-3, 6) and determine the shape of the curve based on the equation y = 2x² + 4x. The tangent line at the point P will have a slope of -8 and pass through P(-3, 6).