The section is on concavity and the second derivative test, the question is:
Determine all significant features and sketch a graph:
f(x) = x / x + 2
To sketch the graph of the function f(x) = x / (x + 2) and determine its significant features, we need to follow these steps:
Step 1: Find the domain of the function.
To determine the domain of f(x), we need to find the values of x for which the function is defined. In this case, the expression in the denominator (x + 2) should not be equal to zero since division by zero is undefined. So, x cannot be equal to -2. The domain of f(x) is all real numbers except -2.
Step 2: Find the x and y-intercepts.
To find the x-intercept, we set f(x) = 0 and solve for x:
0 = x / (x + 2)
Since the numerator is zero, the only solution is x = 0. Therefore, the x-intercept is (0, 0).
To find the y-intercept, we set x = 0 in the function:
f(0) = 0 / (0 + 2) = 0/2 = 0.
Therefore, the y-intercept is (0, 0).
Step 3: Find any vertical or horizontal asymptotes.
A vertical asymptote occurs when the denominator is zero but the numerator is not zero. In our case, the denominator is (x + 2), which is never zero for any real value of x except -2. Therefore, there are no vertical asymptotes.
To find any horizontal asymptotes, we examine the behavior of the function as x approaches positive or negative infinity. Taking the limit as x approaches infinity:
lim(x→∞) f(x) = lim(x→∞) (x / (x + 2))
Applying L'Hôpital's rule by differentiating the numerator and denominator:
lim(x→∞) (1 / 1) = 1.
Similarly, taking the limit as x approaches negative infinity:
lim(x→-∞) f(x) = lim(x→-∞) (x / (x + 2)) = lim(x→-∞) (1 / 1) = 1.
Therefore, the horizontal asymptote for f(x) is y = 1.
Step 4: Find critical points and determine concavity.
To find the critical points of f(x), we need to set the derivative equal to zero and solve for x:
f'(x) = 0
To find the derivative, we can use the quotient rule:
f'(x) = (1*(x + 2) - x*1) / (x + 2)^2 = (2 - x) / (x + 2)^2.
Setting the numerator equal to zero:
2 - x = 0
x = 2.
So, the critical point is x = 2.
Now, we need to determine the concavity of the function. To do this, we find the second derivative and evaluate it at the critical point:
f''(x) = (d^2/dx^2) [(2 - x) / (x + 2)^2]
Using the quotient rule and simplifying, we get:
f''(x) = 6 / (x + 2)^3.
Evaluating f''(2):
f''(2) = 6 / (2 + 2)^3 = 6 / 64 = 3 / 32.
Since f''(2) is positive, the function is concave up at x = 2.
Step 5: Sketch the graph.
Based on the information we have gathered, we can sketch the graph of the function f(x) = x / (x + 2):
- The graph has a vertical asymptote at x = -2.
- There is a horizontal asymptote at y = 1.
- The critical point (or turning point) is x = 2, where the function changes concavity from concave up to concave down.
- The graph intersects the x-axis at (0, 0) and the y-axis at (0, 0).
Combining all these details, we can plot the points and sketch the graph of f(x) = x / (x + 2).