Hello there, assistance would be terrific, thank you very much.
Consider the function f(x)= x^3 + 2x^2 + bx.
a) The equation of the tangent line to the graph of this function at x = 1 is given by
y = ?
b) The tangent line intersects the x-axis at x = ?
c) For what value(s) of b is there no intersection? Why not?
y = x^3+2x^2+bx
y' = 3x^2+4x+b
y'(1) = 3+4+b = 7+b
y(1) = 1+2+b = 3+b
So the line through (1,3+b) with slope 7+b is
y-(3+b) = (7+b)(x-1)
y = (7+b)x - 4
y=0 at x = 4/(7+b)
So, under what condition will the tangent line(s) not intersect the x-axis?
See an example for b = -3:
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E3%2B2x%5E2-3x,+y%3D4x-4
Great, thank you Steve!
Hello! I would be happy to assist you with your questions.
a) To find the equation of the tangent line to the graph of the function at x = 1, we need to find the slope of the tangent line and a point on the line. The slope of the tangent line at a given point can be found by taking the derivative of the function at that point. So, let's take the derivative of f(x) to find the slope at x = 1.
The derivative of f(x) = x^3 + 2x^2 + bx can be found using the power rule for differentiation. The power rule states that if you have a term of the form x^n, its derivative is nx^(n-1).
So, derivative of f(x) = 3x^2 + 4x + b.
Now, we can find the slope of the tangent line at x = 1 by plugging x = 1 into the derivative:
Slope at x = 1 = 3(1)^2 + 4(1) + b
= 3 + 4 + b
= 7 + b.
Now that we have the slope, let's find a point on the line. We can do this by plugging x = 1 into the original function f(x):
f(1) = (1)^3 + 2(1)^2 + b
= 1 + 2 + b
= 3 + b.
So, we have a point (1, 3 + b) on the tangent line. Using the point-slope form of a linear equation, we can write the equation of the tangent line as:
y - (3 + b) = (7 + b)(x - 1).
b) To find where the tangent line intersects the x-axis, we need to find the x-coordinate where the y-value is equal to zero (since that's where the line crosses). So, let's set y = 0 and solve for x:
0 - (3 + b) = (7 + b)(x - 1).
Simplifying this equation will give us the value of x:
-3 - b = (7 + b)(x - 1).
c) To find the value(s) of b for which there is no intersection, we need to find when the equation in part b) has no real solutions for x.
If there are no real solutions for x, it means that the discriminant of the quadratic equation (7 + b)(x - 1) - (-3 - b) = 0 is negative. The discriminant is given by b^2 - 4ac, where a = (7 + b), b = (x - 1), and c = (-3 - b).
Setting the discriminant less than zero, we have:
b^2 - 4ac < 0.
Now, you can solve this inequality for b to find the values for which there is no intersection.
I hope this helps! Let me know if you have any further questions.