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Math  /  Calculus

QuestionGiven the function ff and point a below, complete parts (a)-(c). f(x)=7−6x,a=16f(x)=7-6 x, a=\frac{1}{6}
1 (x) - 6 b. Graph f(x)f(x) and f−1(x)f^{-1}(x) together. Choose the correct graph below. A. B. C. D. c. Evaluate dfdx\frac{d f}{d x} at x=ax=a and df−1dx\frac{d f^{-1}}{d x} at x=f(a)x=f(a) to show that df−1dx∣x=f(a)=1(df/dx)∣x=a\left.\frac{d f^{-1}}{d x}\right|_{x=f(a)}=\frac{1}{\left.(d f / d x)\right|_{x=a}} dfdx∣x=16=\left.\frac{d f}{d x}\right|_{x=\frac{1}{6}}= □\square df−1dx∣x=f(16)=\left.\frac{d f^{-1}}{d x}\right|_{x=f\left(\frac{1}{6}\right)}= □\square (Simplify your answers. Use integers or fractions for any numbers in the expressions.)

Studdy Solution

STEP 1

1. We are given the function f(x)=7−6x f(x) = 7 - 6x .
2. We are given the point a=16 a = \frac{1}{6} .
3. We need to complete parts (a)-(c) as described.

STEP 2

1. Find the inverse function f−1(x) f^{-1}(x) .
2. Graph f(x) f(x) and f−1(x) f^{-1}(x) together and choose the correct graph.
3. Evaluate dfdx\frac{d f}{d x} at x=a x = a .
4. Evaluate df−1dx\frac{d f^{-1}}{d x} at x=f(a) x = f(a) .
5. Verify the relationship between the derivatives of f f and f−1 f^{-1} .

STEP 3

To find the inverse function f−1(x) f^{-1}(x) , start by setting y=f(x) y = f(x) :
y=7−6x y = 7 - 6x
Solve for x x in terms of y y :
y=7−6x y = 7 - 6x 6x=7−y 6x = 7 - y x=7−y6 x = \frac{7 - y}{6}
Thus, the inverse function is:
f−1(x)=7−x6 f^{-1}(x) = \frac{7 - x}{6}

STEP 4

Graph f(x)=7−6x f(x) = 7 - 6x and f−1(x)=7−x6 f^{-1}(x) = \frac{7 - x}{6} together. Choose the correct graph from options A, B, C, or D.
(Note: Since I cannot display graphs, you will need to plot these functions and choose the correct graph based on their characteristics. The graph of f(x) f(x) is a line with a negative slope, and the graph of f−1(x) f^{-1}(x) is a line with a positive slope. They should be reflections of each other across the line y=x y = x .)

STEP 5

Evaluate dfdx\frac{d f}{d x} at x=a x = a .
First, find the derivative of f(x) f(x) :
dfdx=−6 \frac{d f}{d x} = -6
Evaluate at x=16 x = \frac{1}{6} :
dfdx∣x=16=−6 \left. \frac{d f}{d x} \right|_{x = \frac{1}{6}} = -6

STEP 6

Evaluate df−1dx\frac{d f^{-1}}{d x} at x=f(a) x = f(a) .
First, find f(a) f(a) :
f(16)=7−6(16)=7−1=6 f\left(\frac{1}{6}\right) = 7 - 6\left(\frac{1}{6}\right) = 7 - 1 = 6
Now, find the derivative of f−1(x) f^{-1}(x) :
df−1dx=−16 \frac{d f^{-1}}{d x} = -\frac{1}{6}
Evaluate at x=f(a)=6 x = f(a) = 6 :
df−1dx∣x=6=−16 \left. \frac{d f^{-1}}{d x} \right|_{x = 6} = -\frac{1}{6}

STEP 7

Verify the relationship:
df−1dx∣x=f(a)=1dfdx∣x=a \left. \frac{d f^{-1}}{d x} \right|_{x = f(a)} = \frac{1}{\left. \frac{d f}{d x} \right|_{x = a}}
Substitute the values:
−16=1−6 -\frac{1}{6} = \frac{1}{-6}
The relationship holds true.
The values are:
dfdx∣x=16=−6 \left. \frac{d f}{d x} \right|_{x = \frac{1}{6}} = -6
df−1dx∣x=6=−16 \left. \frac{d f^{-1}}{d x} \right|_{x = 6} = -\frac{1}{6}

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