Reposting b/c I'm not sure if anyone will see considering the system thinks I answered it myself:
The graph of the function f is shown below with domain [-12, 10]. Let g(x)=integral[-12,x] f(t)dt.
desmos calculator code: calculator/jycetemxr2 (add desmos url)
If you can't get to the graph, the points are (-12, 5), (-10, 5), (0, -5), (2, -5), (6, 5), and (10, 5). These points are connected with straight lines only, in order.
A. Evaluate g(0), g'(0), and g''(0).
B. Find all possible values of x such that g(x) = 0. Justify your answer.
C. Determine where the graph of g(x) has horizontal tangent lines, if anywhere. Justify your answer.
To answer these questions, we need to understand the properties of the integral function and its relationship with the original function f(x).
First, let's understand the integral function g(x) = ∫[-12,x] f(t) dt. This represents the area under the curve of the function f(x) from -12 to x.
A. To evaluate g(0), we need to find the area under the curve from -12 to 0. Looking at the graph provided, the area is a triangle with a base of length 12 and height 5. Therefore, g(0) = (1/2) * 12 * 5 = 30.
Now, let's find g'(x), which represents the derivative of g(x) with respect to x. Since the integral is a fundamental concept in calculus, we can use the Fundamental Theorem of Calculus to evaluate g'(x). According to this theorem, if F(x) is an antiderivative of f(x), then the derivative of the integral g(x) is simply f(x). In other words, g'(x) = f(x).
B. To find all possible values of x such that g(x) = 0, we need to identify the points on the graph where the area under the curve is zero. From the given graph, we see that the areas above and below the x-axis are equal. These correspond to the intervals (-12, -10) and (2, 6). Therefore, the values of x that satisfy g(x) = 0 are -10 ≤ x ≤ -12 and 2 ≤ x ≤ 6.
C. To determine where the graph of g(x) has horizontal tangent lines, we need to find the values of x where g'(x) = 0. Since g'(x) = f(x), we need to identify the points on the graph where f(x) is zero. Looking at the graph, we see that f(x) is zero at x = -10 and x = 2.
At these points, the graph of g(x) will have horizontal tangent lines.
To summarize:
A. g(0) = 30, g'(0) = f(0) (determined by the graph), g''(0) is not given but can be found by taking the derivative of f(x) and evaluating it at x = 0.
B. x = -10 ≤ x ≤ -12 and 2 ≤ x ≤ 6 are the values for which g(x) = 0.
C. The graph of g(x) has horizontal tangent lines at x = -10 and x = 2.