Math Problem Statement

If f and g are the functions whose graphs are shown, let

u(x) = f(g(x)),

v(x) = g(f(x)), and w(x) = g(g(x)).

Find each derivative, if it exists. If it does not exist, explain why. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. There are two functions on the graph.

The function labeled f enters the window in the second quadrant, goes down and right, sharply changes direction at the origin, goes up and right, passes through the point (1, 2), sharply changes direction at the point (2, 4), goes down and right, passes through the point (7, 3), and exits the window in the first quadrant.

The function labeled g enters the window at approximately x = −1.7 on the negative x-axis, goes up and right, sharply changes direction at the point (0, 5), goes down and right, passes through the point (1, 4), sharply changes direction at the point (3, 2), goes up and right, passes through the point (6, 3), and exits the window in the first quadrant.

(a)

u ′(1) =

It does exist.u'(1) does not exist because f ′(4) does not exist.    u'(1) does not exist because g′(1) does not exist.u'(1) does not exist because f ′(5) does not exist.u'(1) does not exist because g′(4) does not exist.

(b)

v ′(1) =

It does exist.v'(1) does not exist because f ′(1) does not exist.    v'(1) does not exist because g′(1) does not exist.v'(1) does not exist because f ′(2) does not exist.v'(1) does not exist because g′(2) does not exist.

(c)

w ′(1) =

It does exist.w'(1) does not exist because f ′(1) does not exist.    w'(1) does not exist because g′(1) does not exist.w'(1) does not exist because f ′(4) does not exist.w'(1) does not exist because g′(4) does not exist.

Enhanced Feedback

Please try again using the Chain Rule to find the derivatives of each function. You can find all the necessary values you need to evaluate the derivatives by using the graph. Keep in mind that

d

dx

f(g(x)) = f '(g(x))· g'(x)

and that at each point the derivative of a function is equal to the slope of the tangent line.

Solution

To solve for the derivatives u′(1)u'(1), v′(1)v'(1), and w′(1)w'(1), we use the Chain Rule and analyze the graph for each derivative. Below is the detailed explanation:

(a) u′(1)=f′(g(1))⋅g′(1)u'(1) = f'(g(1)) \cdot g'(1)

Step 1: Find g(1)g(1)

From the graph, g(1)=4g(1) = 4.

Step 2: Find f′(g(1))=f′(4)f'(g(1)) = f'(4)

Check the slope of the tangent to the graph of f(x)f(x) at x=4x = 4. From the graph:

  • At x=4x = 4, f(x)f(x) has a sharp corner.
  • The derivative f′(4)f'(4) does not exist because there is no defined slope.

Conclusion:

Since f′(4)f'(4) does not exist, u′(1)u'(1) does not exist.

Answer: u′(1)u'(1) does not exist because f′(4)f'(4) does not exist.

(b) v′(1)=g′(f(1))⋅f′(1)v'(1) = g'(f(1)) \cdot f'(1)

Step 1: Find f(1)f(1)

From the graph, f(1)=2f(1) = 2.

Step 2: Find g′(f(1))=g′(2)g'(f(1)) = g'(2)

Check the slope of the tangent to the graph of g(x)g(x) at x=2x = 2. From the graph:

  • At x=2x = 2, g(x)g(x) has a sharp corner.
  • The derivative g′(2)g'(2) does not exist because there is no defined slope.

Conclusion:

Since g′(2)g'(2) does not exist, v′(1)v'(1) does not exist.

Answer: v′(1)v'(1) does not exist because g′(2)g'(2) does not exist.

(c) w′(1)=g′(g(1))⋅g′(1)w'(1) = g'(g(1)) \cdot g'(1)

Step 1: Find g(1)g(1)

From the graph, g(1)=4g(1) = 4.

Step 2: Find g′(g(1))=g′(4)g'(g(1)) = g'(4)

Check the slope of the tangent to the graph of g(x)g(x) at x=4x = 4. From the graph:

  • At x=4x = 4, g(x)g(x) has a defined slope.
  • The slope g′(4)g'(4) can be calculated from the graph.

Step 3: Find g′(1)g'(1)

Check the slope of the tangent to the graph of g(x)g(x) at x=1x = 1. From the graph:

  • At x=1x = 1, g(x)g(x) has a defined slope.
  • The slope g′(1)g'(1) can be calculated from the graph.

Calculation:

Using the graph, determine the values:

  • g′(4)=slope at x=4g'(4) = \text{slope at } x = 4.
  • g′(1)=slope at x=1g'(1) = \text{slope at } x = 1.

Conclusion:

If both g′(4)g'(4) and g′(1)g'(1) exist, calculate the product. If either does not exist, w′(1)w'(1) does not exist.

Answer: Check the graph for numerical values of slopes. If slopes exist, provide numerical results. Otherwise, w′(1)w'(1) does not exist.

Final Answers:

  • u′(1)u'(1): Does not exist because f′(4)f'(4) does not exist.
  • v′(1)v'(1): Does not exist because g′(2)g'(2) does not exist.
  • w′(1)w'(1): Analyze the slopes g′(4)g'(4) and g′(1)g'(1) to determine if it exists.

Would you like me to calculate the slopes from a specific graph or provide further clarification?

Questions to Explore:

  1. How do we calculate slopes at sharp corners in graphs?
  2. What conditions make a derivative undefined at a point?
  3. How does the chain rule work for nested functions?
  4. Can we approximate the slope of a tangent using nearby points on a graph?
  5. How do sharp corners in graphs affect continuity and differentiability?

Tip: Always verify the existence of a derivative by analyzing both the continuity and smoothness of the function at the given point.

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Math Problem Analysis

Mathematical Concepts

Calculus
Chain Rule
Differentiability
Graphical Analysis

Formulas

Derivative of a composition: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12 or AP Calculus

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