4 Derivatives of Inverse Functions.notebook · 2019-11-18 ·...
12
4_Derivatives_of_Inverse_Functions.notebook 1 November 18, 2019 Derivatives of Inverse Functions Lesson objectives Teachers' notes 1. Apply the properties of inverse functions to the derivatives of inverse functions 2. Apply rules of derivatives for inverse trig functions, exponential and logarithmic functions. Topic 3.3: Differentiating Inverse Functions FUN3: Recognizing opportunities to apply derivative rules can simplify differentiation. FUN3.E: Calculate derivatives of inverse and inverse trigonometric functions. FUN3.E.1: The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists. Teachers' notes Lesson objectives Subject: Topic: Grade(s): Prior knowledge: Crosscurricular link(s): Type text here Type text here Type text here Type text here Type text here Lesson notes: Type text here
Transcript of 4 Derivatives of Inverse Functions.notebook · 2019-11-18 ·...
4_Derivatives_of_Inverse_Functions.notebook
1
November 18, 2019
Derivatives of Inverse Functions
Lesson objectives Teachers' notes
1. Apply the properties of inverse functions to the derivatives of inverse functions
2. Apply rules of derivatives for inverse trig functions, exponential and logarithmic functions.
Topic 3.3: Differentiating Inverse FunctionsFUN3: Recognizing opportunities to apply derivative rules can simplify differentiation.FUN3.E: Calculate derivatives of inverse and inverse trigonometric functions.FUN3.E.1: The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists.
Teachers' notesLesson objectives
Subject:
Topic:
Grade(s):
Prior knowledge:
Crosscurricular link(s):
Type text here
Type text here
Type text here
Type text here
Type text here
Lesson notes:
Type text here
4_Derivatives_of_Inverse_Functions.notebook
2
November 18, 2019
Recall from previous courses that a function, y = f (x), that is one‑to‑one will pass the horizontal line test and will therefore, have a unique inverse function . We often say that the inverse function “undoes” the original function f(x). This can be seen in the test to prove two functions are inverses of each other, since the composition of the two functions produces the identity function.
Derivatives of Inverse Functions
Topic 3.3: Differentiating Inverse Functions
A function g is the inverse function of the for each x in the
The function g is denoted by domain of f.
Definition of an Inverse:
domain of g and for each x in the.
function f if
Reflective Property of Inverse Functions:
The graph of f contains the point if and only if the graph of
contains the point .
4_Derivatives_of_Inverse_Functions.notebook
3
November 18, 2019
EX #1: Use the reflective property of inverse functions and the function values for f(x) and g(x) shown in the table below, to complete the table of values for their respective inverse functions, and . Then, use them to answer each of the questions below.
A. Complete the table of values for :
B. Complete the table of values for :
C. Find the value of D. Find the value of
E. Find the value of F. Find the value of
4_Derivatives_of_Inverse_Functions.notebook
4
November 18, 2019
THE EXISTENCE OF AN INVERSE FUNCTION:
1. A function has an inverse function if and only if it is onetoone.
2. If f is strictly monotonic on its entire domain, (either increasing or decreasing on its entire domain) then it is onetoone and therefore has an inverse function.
The relationship between a composite function and its inverse is stated below. If you use the chain rule and differentiate both sides, you can find a formula for the derivative of an inverse.
4_Derivatives_of_Inverse_Functions.notebook
5
November 18, 2019
EX #2: Given and Find the value of
GOAL: How do we find the derivative of the inverse function of f(x) at a point x = a?
Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable atany x for which Moreover,
THE DERIVATIVE OF AN INVERSE FUNCTION:
4_Derivatives_of_Inverse_Functions.notebook
6
November 18, 2019
To find for a point (a, b) on f(x).
4.
1. Find
2. If you are only given b, set b = f(x) to find a.
3. Find
To Find the Derivative of the Inverse Function:
Note:
Let f and g be inverse functions, such that where and
*** Inverse functions have reciprocal slopes at corresponding points.***
EX #3: Find where g(x) is the inverse of f(x)
4_Derivatives_of_Inverse_Functions.notebook
7
November 18, 2019
CONTINUITY AND DIFFERENTIABILITYOF INVERSE FUNCTIONS
Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true.
1. If f is continuous on its domain, then is continuous on its domain.
is increasing on its domain.2. If f is increasing on its domain, then
3. If f is decreasing on its domain, thenis decreasing on its domain.
4. If f is differentiable at c andthen is differentiable at
EX #4: Let g be the inverse of f. Calculate for
4_Derivatives_of_Inverse_Functions.notebook
8
November 18, 2019
There are questions where you will need to use a calculator to find approximations accurate to three decimal places. Here’s an example of such a task.
EX #5: If , find accurate to three decimal places.
1. Solve f(x) = 3, this will give you x = a.2. Differentiate f(x)3. Evaluate f ′ (a) at the value from step 1, using all the decimal
values, no early rounding.4. Use your result in the inverse derivative formula
Derivatives of Other Bases:Let a be a positive real number (a ≠ 1) and let u be a
differentiable function of x.
1. 2.
3. 4.
4_Derivatives_of_Inverse_Functions.notebook
9
November 18, 2019
A.) B.)
C.) D.)
EX #6: Find the derivative of each of the following.
EX #7: Find the derivative of each of the following.
A.) B.)
C.) D.)
4_Derivatives_of_Inverse_Functions.notebook
10
November 18, 2019
EX #8: Find an equation of the tangent line of where x = 2.
EX # 9: Bacterial Culture Growth
where W(t) is the weight of the culture in ounces and t is in hours. Find the weight of the culture after 0 hour, 2 hours, 8 hours. What is the limit as t approaches infinity?
A bacterial culture is growing according to the function
4_Derivatives_of_Inverse_Functions.notebook
11
November 18, 2019
EX #10: Inverse Derivatives by Tables
Use the table below to find each of the indicated values below. State the ordered pairs for the inverse function in the column provided.
A. Find B. Find
C. Write the equation for the line tangent to the graph of when x = 2.
4_Derivatives_of_Inverse_Functions.notebook
12
November 18, 2019
D. Estimate the value of . Use the result to explain the behavior of the graph of the function g when x = 3.
