Derivatives can be calculated from graphed functions.
The values of the derivatives are:
The given parameter is:
[tex]\mathbf{h(x) = f(g(x))}[/tex]
Start by calculating the equations of f(x) and g(x)
Graph f(x)
The slopes of f(x) are: 3/2 and -3/2
So, the equations are:
[tex]\mathbf{f(x) = \frac{3}{2}x,\ 0 \le x \le 2}[/tex]
[tex]\mathbf{f(x) = -\frac{3}{2}x,\ x \ge 2}[/tex]
Graph g(x)
The slope of g(x) is: -1/2
So, the equation is:
[tex]\mathbf{g(x) = -\frac 12x}[/tex]
For x = 1 and x = 2, we have:
So, we have:
[tex]\mathbf{h(x) = f(g(x))}[/tex]
Where:
[tex]\mathbf{f(x) = \frac{3}{2}x\ 0 \le x \le 2}[/tex] and [tex]\mathbf{g(x) = -\frac 12x}[/tex]
[tex]\mathbf{h(x) = f(g(x))}[/tex] becomes
[tex]\mathbf{h(x) = \frac{3}{2}(-\frac{1}{2}x)}[/tex]
Open brackets
[tex]\mathbf{h(x) = -\frac{3}{4}x}[/tex]
Differentiate
[tex]\mathbf{h'(x) = -\frac{3}{4}}[/tex]
So:
[tex]\mathbf{h'(1) = -\frac{3}{4}}[/tex]
[tex]\mathbf{h'(2) = -\frac{3}{4}}[/tex]
For x = 3, we have:
[tex]\mathbf{h(x) = f(g(x))}[/tex]
Where:
[tex]\mathbf{f(x) = -\frac{3}{2}x\ x \ge 2}[/tex] and [tex]\mathbf{g(x) = -\frac 12x}[/tex]
[tex]\mathbf{h(x) = f(g(x))}[/tex] becomes
[tex]\mathbf{h(x) = -\frac{3}{2}(-\frac{1}{2}x)}[/tex]
[tex]\mathbf{h(x) = \frac{3}{4}x}[/tex]
Differentiate
[tex]\mathbf{h'(x) = \frac{3}{4}}[/tex]
Substitute 3 for x
[tex]\mathbf{h'(3) = \frac{3}{4}}[/tex]
Hence, the values of the derivatives are:
[tex]\mathbf{h'(1) = -\frac{3}{4}}[/tex], [tex]\mathbf{h'(2) = -\frac{3}{4}}[/tex] and [tex]\mathbf{h'(3) = \frac{3}{4}}[/tex]
Read more about graphed functions at:
https://brainly.com/question/11804653
