17Calculus Precalculus - Exponential Functions
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What Are Exponential Functions? |
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Exponentials are very similar to polynomials and are easily confused. Check out the next two examples.
\(f(x)=x^2\) |
\(g(x)=2^x\) | |
exponential |
The polynomial you should be familiar with. Notice that the difference between a polynomial and an exponential is where the variable is, in the base or in the exponent. In the base, we have a polynomial, in the exponent, we have an exponential function.
In general, an exponential function is written \(f(x) = Ca^x\) where a is a positive real number (\(a \gt 0\)) and \(a \neq 1 \) and C is a real number.
a is called the growth factor
C is called the initial value since \(f(0) = Ca^0 = C\)
Remember these laws of exponents? These also hold for exponential functions.
1. |
\(a^s \cdot a^t = a^{s+t}\) |
2. |
\((a^s)^t = a^{st}\) |
3. |
\( (ab)^s = a^s \cdot b^s \) |
4. |
\( 1^s = 1 \) |
5. |
\(\displaystyle{ a^{-s} = \frac{1}{a^s} = \left[ \frac{1}{a} \right]^s }\) |
6. |
\( a^0 = 1 \) since \(a\neq0\) |
7. |
if \(a^u = a^v\) then \(u=v\) |
Also remember that \((a+b)^s \neq a^s + b^s\). An example of this is \((a+b)^2 = a^2 +2ab + b^2 \neq a^2 + b^2\).
Before we go on, here is a good video that shows several examples so that you can get a feel for exponential functions.
PatrickJMT - Identify the Exponential Function [2min-34secs]
video by PatrickJMT |
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Notice we are calling \(g(x)=a^x\) a function. The use of the word 'function' is deliberate since exponentials pass the vertical line test. Let's look at a few exponential functions. We will draw them on the same graph so that we can compare them.
In plot 1 below, notice that all of the exponential functions have the same general shape and that they all go through the point \((0,1)\). This makes sense, since \(2^0=3^0=4^0=1\). This shape holds for exponential functions \(y=a^x\) where \(a>1\).
Now take a careful look at plot 2. Notice that the exponential functions are still in the form \(y=a^x\) but in these cases \(0 < a < 1\). Said another way, \(a > 1\) and \(y=(1/a)^x\). And said still another way, \(a > 1\) and \(y=a^{-x}\).
Again, all plots go through the point \((0,1)\) and they all have the same general shape.
When comparing the two plots, notice that they are flipped with respect to the y-axis. These are the kind of things you need to look at and think about in calculus when you are given a graph.
plot 1: \(\color{red}{y=2^x}\); \(\color{blue}{y=3^x}\); \(\color{green}{y=4^x}\) |
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plot 2: \(\color{red}{y=(1/2)^x}\); \(\color{blue}{y=(1/3)^x}\); \(\color{green}{y=(1/4)^x}\) |
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Study Tip |
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When given alternate ways of describing an equation like we have done here, focus on the one that makes most sense to you and then determine the relationship between the one you chose and the other ones. Learning more than one at time can be overwhelming and an unnecessary waste of your time and mental energy. |
Okay, let's watch a video clip that explains this in more detail.
MIP4U - Graph Exponential Functions [7min-55secs]
video by MIP4U |
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Time for some practice problems.
Practice
Unless otherwise instructed, graph these exponential functions.
Sketch the graph of \( y=2^x \) and find the domain and range.
Problem Statement |
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Sketch the graph of \( y=2^x \) and find the domain and range.
Solution |
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2987 video
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Sketch the graph of \(y=(1/3)^x \) and find the domain and range.
Problem Statement |
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Sketch the graph of \(y=(1/3)^x \) and find the domain and range.
Solution |
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2988 video
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Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.
Problem Statement |
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Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.
Solution |
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2989 video
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Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.
Problem Statement |
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Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.
Solution |
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2990 video
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Unless otherwise instructed, graph the exponential function \(y=2^x+1\).
Problem Statement |
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Unless otherwise instructed, graph the exponential function \(y=2^x+1\).
Solution |
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1693 video
video by PatrickJMT |
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Unless otherwise instructed, graph the exponential function \(y=3^{-x}-2\).
Problem Statement |
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Unless otherwise instructed, graph the exponential function \(y=3^{-x}-2\).
Solution |
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1694 video
video by PatrickJMT |
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Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).
Problem Statement |
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Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).
Solution |
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1695 video
video by PatrickJMT |
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Unless otherwise instructed, graph the exponential function \(y=-3^x-1\).
Problem Statement |
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Unless otherwise instructed, graph the exponential function \(y=-3^x-1\).
Solution |
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1696 video
video by PatrickJMT |
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Determine the equation of this transformed exponential function.
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Problem Statement |
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Determine the equation of this transformed exponential function.
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Solution |
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1697 video
video by MIP4U |
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Unless otherwise instructed, graph the exponential function \(y=5^x\).
Problem Statement |
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Unless otherwise instructed, graph the exponential function \(y=5^x\).
Solution |
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1690 video
video by Khan Academy |
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Special Exponential Function |
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There is a special exponential function that is used extensively in calculus, \(f(x)=e^x\). The number \(e \approx 2.7183\) is a special irrational number. (You need calculus to understand where it comes from.) Since \(e > 1\), the graph looks like plot 1 above.
Solving Exponential Equations |
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In order to get a feel for working with exponential functions, you may be asked to solve exponential functions. The first technique does not require the knowledge of logarithms. This next video contains an explanation how to do this followed by lots of examples.
MIP4U - Solving Exponential Equations [6min-27secs]
video by MIP4U |
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Practice
Solve these exponential functions.
Solve \( 3^{x+2} = 9^{2x-3} \)
Problem Statement |
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Solve \( 3^{x+2} = 9^{2x-3} \)
Solution |
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2991 video
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Solve \( 8^{4x-12} = 16^{5x-3} \)
Problem Statement |
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Solve \( 8^{4x-12} = 16^{5x-3} \)
Solution |
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2992 video
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Solve \( 27^{3x-2} = 81^{2x+7} \)
Problem Statement |
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Solve \( 27^{3x-2} = 81^{2x+7} \)
Solution |
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2993 video
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Solve \( 3^x = 8 \)
Problem Statement |
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Solve \( 3^x = 8 \)
Solution |
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2994 video
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Solve \( e^x = 7 \)
Problem Statement |
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Solve \( e^x = 7 \)
Solution |
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2995 video
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Solve \( 5 + 4^{x-2} = 23 \)
Problem Statement |
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Solve \( 5 + 4^{x-2} = 23 \)
Solution |
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2996 video
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Solve \( 3 + 2e^{3-x} = 7 \)
Problem Statement |
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Solve \( 3 + 2e^{3-x} = 7 \)
Solution |
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2997 video
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Solve \( 3^{x^2+4x} = 1/27 \)
Problem Statement |
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Solve \( 3^{x^2+4x} = 1/27 \)
Solution |
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2998 video
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Solve \( 2^{x^2} \cdot 2^{3x} = 16 \)
Problem Statement |
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Solve \( 2^{x^2} \cdot 2^{3x} = 16 \)
Solution |
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2999 video
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Solve \( 4^{2x} - 20 \cdot 4^x+ 64 = 0 \)
Problem Statement |
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Solve \( 4^{2x} - 20 \cdot 4^x+ 64 = 0 \)
Solution |
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3000 video
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Solve \( 3^{2x} - 3^{2x-1} = 18 \)
Problem Statement |
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Solve \( 3^{2x} - 3^{2x-1} = 18 \)
Solution |
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3001 video
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Unless otherwise instructed, solve the exponential function \(3^{2x+5}=9^{3x-7}\).
Problem Statement |
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Unless otherwise instructed, solve the exponential function \(3^{2x+5}=9^{3x-7}\).
Solution |
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1691 video
video by MIP4U |
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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \frac{1}{27}=9^{5x-7} }\).
Problem Statement |
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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \frac{1}{27}=9^{5x-7} }\).
Solution |
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1689 video
video by PatrickJMT |
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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\).
Problem Statement |
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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\).
Solution |
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1692 video
video by MIP4U |
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Unless otherwise instructed, solve the exponential function \(3^x + 2\cdot 3^{x+1} = 21\).
Problem Statement |
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Unless otherwise instructed, solve the exponential function \(3^x + 2\cdot 3^{x+1} = 21\).
Solution |
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2127 video
video by Dr Chris Tisdell |
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Rules of Exponents - Review |
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Here are some videos to help you review rules of exponents. These videos also contain plenty of examples.
PatrickJMT - Basic Exponent Properties [12min-26secs]
video by PatrickJMT |
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MIP4U - Simplify Exponential Expressions With Negative Exponents - Basic [4min-44secs]
video by MIP4U |
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Next |
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Exponentials are very closely related to logarithms. So, after working the practice problems on this page, your next logical step is logarithms. After that, we will discuss how to use exponentials and logarithms together and then solve some application problems.
exponentials 17calculus youtube playlist
Here is a playlist of the videos on this page.
Really UNDERSTAND Precalculus
Topics You Need To Understand For This Page
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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