The function E, where E(x)=2^x, is an exponential function. (a) Find the values of E(1), E(2) an

The function E, where E(x)=2^x, is an exponential function....

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Key Concepts

Exponential Functions

Exponential functions are mathematical functions in which a constant base is raised to a variable exponent, typically written in the form f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay depending on whether the base is greater than 1 or between 0 and 1, and they are widely used in modeling real-world phenomena such as population growth and radioactive decay.

Evaluating Function Values

Evaluating an exponential function involves substituting a given value for the variable into the function and computing the result using the properties of exponents. This process demonstrates the direct application of arithmetic operations on exponents, helping in understanding the behavior and output of the function at specific points.

Solving Exponential Equations

Solving exponential equations generally entails finding the value of the exponent variable that satisfies the equation. This can sometimes be done by expressing both sides of the equation with the same base, thereby allowing the exponents to be equated directly, or by using logarithms when such a simplification is not possible. The technique emphasizes the inverse relationship between exponential functions and logarithms, which is central to many areas of algebra.

Transcript

00:01 Since there's a lot of problems to do, i'm just going to write down the notation that i'm finding the derivative of 2 to the 3x.

00:10 And this is pretty straightforward, especially following exponential, that it's just natural log of the base times you leave the function alone, 2 to the 3x.

00:22 But with the chain rule, you have to multiply by the derivative of the exponent, which is 3.

00:27 And this is a good answer.

00:28 You can just stop right there.

00:30 So if you look at letter b, where we're asked to find the derivative of 3 .1 to the x.

00:40 So there's your exponential.

00:42 And then plus x cubed, that's actually going to be the power rule.

00:48 So, you know, the exponential, you do natural log of 3 .1, and you multiply by 3 .1 to the x.

00:56 And it's not really the chain rule because the derivative of the exponent is just one.

01:00 But then you have this plus.

01:02 Don't forget the power rule, the derivative of x cubed is just 3x squared.

01:08 And that's it for that problem.

01:11 Looking at letter c, we're asked to find the derivative of 10 to the 3t minus 5.

01:21 It's in terms of t now.

01:27 And the x1 is just 3t minus 5.

01:30 But anyway, it's the same idea...

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