Limit of Exponential Functions
Christian Parkinson GRE Prep: Calculus I Practice Problem Solutions 3 so fis constant. Let f(x) = x2+sin(x) for x0. The temptation here is to use the power rule or the exponential rule but in the. In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the.
Definition
A quantity grows linearly over time if it increases by a fixed amount with each time interval. A quantity decreases linearly over time if it decreases by a fixed amount with each time interval.
Example (PageIndex{1}):
If you start with $1000 and put $200 in a jar every month to save for a vacation, then every month the vacation savings grow by $200 and in x months you will have: Amount = 1000 + 200x
Definition
A quantity grows exponentially over time if it increases by a fixed percentage with each time interval. A quantity decays exponentially over time if it decreases by a fixed percentage with each time interval.
Example (PageIndex{2}):
If you start with a debt of $1000 and you are charged an annual interest rate of 24 percent (typical credit card interest rate) then how much will you owe after X months?
1.4 Limits Of Exponential Functionsap Calculus Formula
24 percent per year = 2 percent per month (this is how they convert it to a monthly interest rate)
EXPONENTIAL FUNCTION
For any real number (x), an exponential function is a function with the form
(f(x)=b^x)
where
- (b) is any positive real number such that (b≠1).
- The domain of (f) is all real numbers.
- The range of (f) is all positive real numbers.
- The y-intercept is ((0,1)),
- and the horizontal asymptote is (y=0).
CHARACTERISTICS OF THE EXPONENTIAL FUNCTION
An exponential function with the form (f(x)=b^x), (b>0), (b≠1),has these characteristics:
- one-to-one function
- horizontal asymptote: (y=0)
- domain: ((–infty, infty))
- range: ((0,infty))
- x-intercept: none
- y-intercept: ((0,1))
- increasing if (b>1)
- decreasing if (b<1)
Rule: Laws of Exponents
For any constants (a>0),(b>0), and for all x and y,
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- (b^x⋅b^y=b^{x+y})
- (dfrac{b^x}{b^y}=b^{x−y})
- ((b^x)^y=b^{xy})
- ((ab)^x=a^xb^x)
- (dfrac{a^x}{b^x}=(dfrac{a}{b})^x)
Example (PageIndex{2}): Using the Laws of Exponents
Use the laws of exponents to simplify each of the following expressions.
- (dfrac{(2x^{2/3})^3}{(4x^{−1/3})^2})
- (dfrac{(x^3y^{−1})^2}{(xy^2)^{−2}})
Soution
a. We can simplify as follows:
(dfrac{(2x^{2/3})^3}{(4x^{−1/3})^2})=(dfrac{2^3(x^{2/3})^3}{4^2(x^{−1/3})^2})=(dfrac{8x^2}{16x^{−2/3}})=(dfrac{x^2x^{2/3}}{2})=(dfrac{x^{8/3}}{2}.)
b. We can simplify as follows:
(dfrac{(x^3y^{−1})^2}{(xy^2)^{−2}}=dfrac{(x3)^2(y^{−1})^2}{x−2(y^2)^{−2}}=dfrac{x^6y^{−2}}{x^{−2}y^{−4}} =x^6x^2y^{−2}y^4=x^8y^2).
Exercise (PageIndex{2}):
Exponential Functions Khan Academy
Use the laws of exponents to simplify ((6x^{−3}y^2)/(12x^{−4}y^5)).
(x^a/x^b=x^{a-b})
1.4 Limits Of Exponential Functionsap Calculus Equation
(x/(2y^3))