Understanding Exponential Functions in Calculus II
Dive into the world of exponential functions in Calculus II as we explore the inverse of natural logarithm functions, evaluate exponential functions, and learn the laws governing these mathematical concepts. Discover how to solve equations involving exponentials and differentiate functions to deepen your understanding of this fundamental topic.
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Calculus II Lecture #3 Exponential Functions Exponential Functions ?? Civil Engineering Department College of Engineering Mustansiriayah University May 2020 1 Calculus II, Lecture #3 19/06/2025
Exponential Functions Exponential Functions The inverse of the natural logarithm function ln x is the exponential function: exp ? = ?? The Inverse of ln x and the Number e The function ln x, being an increasing function of x with domain (0, ) and range (- , ), has an inverse ln-1x with domain (- , ) and range (0, ). The graph of ln-1x is the graph of ln x reflected across the line y = x. As you can see in the figure below: x ?? -1 0 1 2 10 100 0.37 1 2.72 7.39 22026 2.688x1043 ? = ln 11 = ?1 ln? = 1, 2 Calculus II, Lecture #3 19/06/2025
Inverse Equations for Inverse Equations for ??and and ln? ?ln ?= ?, ln ??= ?, for all ? > 0 for all ? Example 1: Solve the equation e2x-6= 4 for x, Solution: We take the natural logarithm of both sides of the equation and use the second inverse equation: ln e2x 6= ln4 2x 6 = ln4 x = 3 +1 2ln4 1 2= 3 + ln2 x = 3 + ln4 3 Calculus II, Lecture #3 19/06/2025
Example Example 2 2: By Using the Inverse equation, the following quantities can be found easily: ln?2= 2 ln? 1= 1 ln ? =1 2 ln?sin ?= sin? ?ln 2= 2 ?ln ?2+1= ?2+1 ?3ln 2= ?ln 23= ?ln 8= 8 4 Calculus II, Lecture #3 19/06/2025
Evaluating the Exponential Functions Evaluating the Exponential Functions 23= ?3ln 2 ?1.2 3.32 2?= ? ln 2 ?2.18 8.8 Laws of Exponential Functions Laws of Exponential Functions ??1.??1= ??1+?2 1 ? ?= ?? ??1 ??2= ??1 ?2 ??1 ?2= ??1?2= ??2 ?1 5 Calculus II, Lecture #3 19/06/2025
Example 3: Applying the Exponential Functions ??+ln 2= ??.?ln 2= 2?? ? ln ?=?ln1 ?=1 ? ?2? ?1= ?2? 1 ?3 ?= ?3?= ?? 3 6 Calculus II, Lecture #3 19/06/2025
The derivative of ?? ? = ??= ln 1? = ? 1? ? ??ln 1? 1 ? ??= ? ? = ln? and ?? ??= ? ????= 1 ? ? 1? ? ??? 1? = 1 1 ?? = ?? = In summary: ? ????= ?? ? ????= ???? ?? Example 4: Differentiate the following functions: ? ??5??= 5 ? ??? ?= ? ?? ? ??? sin ?= ? sin ?? ? ????= 5?? ?? ? = ? ? 1 = ? ? ??sin? = ?sin ?cos? 7 Calculus II, Lecture #3 19/06/2025
The Integral of ?? ???? = ??+ ? Example 5: Integrate the following Exponentials: ln 2 ln 2?3??? = 1 3?3? 3?3 ln 2 ?3(0)=1 ?sin ? 0 =1 38 1 =7 1. 0 3 0 ?/2= ?1 ?0= ? 1 ?/2?sin ?cos? ?? = 2. 0 8 Calculus II, Lecture #3 19/06/2025
Example 6: Find dy/dx for the function: ? = ?3.? 2?.cos5? Solution: ln? = ln?3+ ln? 2?+ ln cos5? ln? = 3ln? 2? + ln cos5? 1 ? ?? ??=3 ? 2 5sin5? cos5? ?? ??= ?3.? 2?.cos5? 3 ? 2 5sin5? cos5? 9 Calculus II, Lecture #3 19/06/2025
Example Example 7 7: Find dy/dx for the function: ? = sin ??+ ?ln ? Solution: ? = sin?ln ??+ ?ln ? ? = sin ?xln ? + ? ?? ??= cos ?xln ? ? ?+ ln? + 1 .?xln ?. ?? ??= cos ?xln ? .?xln ?+ cos ?xln ? .?xln ?.ln? + 1 10 Calculus II, Lecture #3 19/06/2025
Homework Simplify the following expressions: 2ln ? ln ? ?2 ?2 ln ln?? Solve for y in terms of x for the following equations: ln 1 2? = ? ln ? 1 ln2 = ? + ln? ln ?2 1 ln ? + 1 = ln sin? ?ln 2 ?=1 2 Find the derivatives of y with respect to x: ? = ??? ?? 4?+1 ? = ? ? ? = ??sin? + cos? ? = ln 3?? ? Evaluate the following integrals: ?3?+ 5? ??? ?sec ??sec?? tan?? ?? ln 16 ??/4?? 0 11 Calculus II, Lecture #3 19/06/2025
