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Learning Objectives. 4.10.1 Find the general antiderivative of a given function. 4.10.2 Explain the terms and notation used for an indefinite integral. 4.10.3 State the power rule for integrals. 4.10.4 Use antidifferentiation to solve simple initial-value problems.
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[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.
In this section, we explore integration involving exponential and logarithmic functions. Integrals of Exponential Functions. The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, y = e x, y = e x, is its own derivative and its own integral.
We answer the first part of this question by defining antiderivatives. The antiderivative of a function \(f\) is a function with a derivative \(f\). Why are we interested in antiderivatives? The need for antiderivatives arises in many situations, and we look at various examples throughout the remainder of the text.
10 lis 2023 · Example \(\PageIndex{2}\): Square Root of an Exponential Function. Find the antiderivative of the exponential function \(e^x\sqrt{1+e^x}\). Solution. First rewrite the problem using a rational exponent: \[∫e^x\sqrt{1+e^x}\,dx=∫e^x(1+e^x)^{1/2}\,dx.\nonumber \] Using substitution, choose \(u=1+e^x\). Then, \(du=e^x\,dx\). We have
Example \(\PageIndex{1}\): Derivative of an Exponential Function. Find the derivative of \(f(x)=e^{\tan(2x)}\). Solution: Using the derivative formula and the chain rule, \[f′(x)=e^{\tan(2x)}\frac{d}{dx}\Big(\tan(2x)\Big)=e^{\tan(2x)}\sec^2(2x)⋅2 \nonumber \]
Learning Objectives. Find the general antiderivative of a given function. Explain the terms and notation used for an indefinite integral. State the power rule for integrals. Use antidifferentiation to solve simple initial-value problems.
