Example Question - exponential function
Here are examples of questions we've helped users solve.
Finding the Inverse of an Exponential Function
<p>Let \( y = e^{4x} - 5 \)</p> <p>Swap x and y to find the inverse: \( x = e^{4y} - 5 \)</p> <p>Add 5 to both sides: \( x + 5 = e^{4y} \)</p> <p>Take the natural logarithm of both sides: \( \ln(x + 5) = \ln(e^{4y}) \)</p> <p>Use the property of logarithms: \( \ln(x + 5) = 4y \)</p> <p>Divide by 4: \( y = \frac{1}{4}\ln(x + 5) \)</p> <p>The inverse function is \( f^{-1}(x) = \frac{1}{4}\ln(x + 5) \)</p>
Solving a Definite Integral Involving Exponential and Power Functions
<p>To solve the integral \(\int_0^{1/2} x^{1/2}e^{2x} \, dx\), we can use the method of integration by parts, which states that \(\int u \, dv = uv - \int v \, du\).</p> <p>Let \( u = x^{1/2} \) and \( dv = e^{2x} \, dx \).</p> <p>Then we have \( du = \frac{1}{2}x^{-1/2} \, dx \) and \( v = \frac{1}{2}e^{2x} \).</p> <p>The integral becomes:</p> <p>\( \int_0^{1/2} x^{1/2}e^{2x} \, dx = \left. \frac{1}{2}x^{1/2}e^{2x} \right|_0^{1/2} - \int_0^{1/2} \frac{1}{2}e^{2x} \frac{1}{2}x^{-1/2} \, dx \)</p> <p>\( = \left. \frac{1}{2}x^{1/2}e^{2x} \right|_0^{1/2} - \frac{1}{4} \int_0^{1/2} x^{-1/2}e^{2x} \, dx \)</p> <p>To solve the remaining integral, we use integration by parts again with \( u = x^{-1/2} \) and \( dv = e^{2x} \, dx \).</p> <p>Then we get \( du = -\frac{1}{2}x^{-3/2}dx \) and \( v = \frac{1}{2}e^{2x} \).</p> <p>The remaining integral is:</p> <p>\( \frac{1}{4} \left( \left. x^{-1/2}e^{2x} \right|_0^{1/2} - \int_0^{1/2} \frac{1}{2}e^{2x}(-\frac{1}{2})x^{-3/2} \, dx \right) \)</p> <p>\(= \frac{1}{4} \left( \left. x^{-1/2}e^{2x} \right|_0^{1/2} + \frac{1}{4} \int_0^{1/2} x^{-3/2}e^{2x} \, dx \right) \)</p> <p>The evaluation of these integrals at the limits \(0\) and \(\frac{1}{2}\) must be approached with caution, because the terms involving \(x^{-1/2}\) and \(x^{-3/2}\) are undefined at \(x=0\). These expressions suggest the integral does not converge at the lower limit, however, this is dependent on the behavior of the exponential function as it approaches zero, which could negate the potential divergence caused by the power of \(x\).</p> <p>Without performing a limit analysis, the solution remains indeterminate at \(x = 0\). A detailed analysis might involve considering the limit of the integrand as \(x\) approaches zero and applying L'Hôpital's rule if necessary.</p>
Finding the Integral of an Exponential Function
<p>\int 7^x dx = \int e^{x \ln(7)} dx</p> <p>Let u = x \ln(7) \Rightarrow du = \ln(7) dx</p> <p>dx = \frac{du}{\ln(7)}</p> <p>\int e^{x \ln(7)} dx = \int e^u \frac{du}{\ln(7)}</p> <p>\frac{1}{\ln(7)}\int e^u du = \frac{1}{\ln(7)} e^u + C</p> <p>\frac{1}{\ln(7)} e^{x \ln(7)} + C = \frac{7^x}{\ln(7)} + C</p> <p>So, the integral of f(x) = 7^x is \frac{7^x}{\ln(7)} + C</p>
Solving an Exponential Function's Integral
<p>\[\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\]</p> <p>where \(C\) is the constant of integration.</p>
Finding the Integral of an Exponential Function
<p>\(\int 2 \cdot 5^x dx\)</p> <p>\(= 2 \int 5^x dx\)</p> <p>\(= 2 \cdot \frac{1}{\ln(5)} 5^x + C\)</p> <p>where \(C\) is the constant of integration.</p>
Finding the Derivative of an Exponential Function
<p>\(\frac{d}{dx}f(x) = \frac{d}{dx}(2 \cdot 5^x)\)</p> <p>\(= 2 \cdot \frac{d}{dx}(5^x)\)</p> <p>\(= 2 \cdot 5^x \ln(5)\)</p> <p>Thus, \(f'(x) = 2 \cdot 5^x \ln(5)\).</p>
Determining the Function Type Based on the Exponential Expression
<p>The function \( f(x) = 2 \cdot 5^x \) is an exponential function because it can be written in the form \( f(x) = ab^x \), where:</p> <p>\( a = 2 \) (the coefficient),</p> <p>\( b = 5 \) (the base), and</p> <p>\( x \) is the exponent.</p>
Determine the Derivative of the Exponential Function
<p>\( f(x) = 5e^x \)</p> <p>\( f'(x) = \frac{d}{dx}(5e^x) \)</p> <p>\( f'(x) = 5\frac{d}{dx}(e^x) \) (Constant multiple rule)</p> <p>\( f'(x) = 5e^x \) (Derivative of \( e^x \) is \( e^x \))</p>
Butterfly Population Growth Calculation
To solve the given function for the population of butterflies after 7 years, plug in the value of x = 7 into the function f(x) and calculate the result: f(x) = 2,000(1.005)^x So for x = 7, f(7) = 2,000(1.005)^7 Now calculate the value: f(7) = 2,000 * (1.005)^7 Using a calculator, raise 1.005 to the 7th power: (1.005)^7 ≈ 1.0355 Then multiply this result by 2,000: f(7) ≈ 2,000 * 1.0355 f(7) ≈ 2,071 The closest answer to 2,071 is about 2,100, so the correct choice would be: B. about 2,100
Butterfly Population Growth Calculation
The given function is \( f(x) = 2,000(1.055)^x \), where \( x \) represents the number of years, and \( f(x) \) represents the population of butterflies. To find the population after 7 years, we substitute \( x \) with 7: \( f(7) = 2,000(1.055)^7 \) Using a calculator, we find: \( f(7) ≈ 2,000(1.455) \) \( f(7) ≈ 2,910 \) The closest answer to 2,910 is about 2,900, so the correct answer is: A) about 2,800
Finding Inverse of Exponential Function
To find the inverse of the exponential function \( y = 4^x \), you need to solve for x in terms of y. Starting with the original function: \[ y = 4^x \] Swap the roles of x and y to begin finding the inverse function: \[ x = 4^y \] Now, solve for y by taking the logarithm with base 4 of both sides (since 4 is the base of the exponential function): \[ \log_4(x) = \log_4(4^y) \] By the properties of logarithms, \( \log_b(b^a) = a \), so: \[ \log_4(x) = y \] Therefore, the inverse function is: \[ y = \log_4(x) \] Looking at the options provided: A. \( y = x^{-4} \) - Incorrect, because this represents a power function, not a logarithmic function. B. \( y = (\frac{1}{4})^x \) - Incorrect, this is another exponential function, not the inverse of \( 4^x \). C. \( y = \log_4(x) \) - Correct, as proved above. D. \( y = - \log_4(x) \) - Incorrect, this is the negative of the logarithmic function. The correct answer is C: \( y = \log_4(x) \).
Inverse of Exponential Function
The question asks for the inverse of the exponential function \( y = 4^x \). To find the inverse function, we typically swap \( x \) and \( y \) and then solve for \( y \). Starting with \( y = 4^x \), we swap \( x \) and \( y \) to get \( x = 4^y \). Now we need to solve for \( y \). To do this, we take the logarithm base 4 of both sides: \( \log_4(x) = \log_4(4^y) \). Using the property of logarithms that \( \log_b(b^a) = a \), we simplify the right side to get: \( \log_4(x) = y \). So, the inverse function of \( y = 4^x \) is \( y = \log_4(x) \). The correct answer is: C. \( y = \log_4(x) \).
Calculating Average Rate of Change for Exponential Function
The question asks for the average rate of change of the function f(x) = 100 * 2^x on the interval [0,4]. The average rate of change of a function over the interval [a, b] is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Here, our function f(x) = 100 * 2^x, our interval is [0, 4], so a = 0 and b = 4. We plug these into the function f to get f(0) and f(4): f(0) = 100 * 2^0 = 100 * 1 = 100 f(4) = 100 * 2^4 = 100 * 16 = 1600 Now plug f(0) and f(4) into the rate of change formula: \[ \text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{1600 - 100}{4} = \frac{1500}{4} = 375 \] Hence, the average rate of change of f(x) on the interval [0,4] is 375.
Solving Exponential Equation with Natural Logarithm
To solve the equation \( e^{4 - 7x} + 11 = 20 \), you first isolate the exponential part: 1. Subtract 11 from both sides to move the constant term to the right-hand side of the equation: \[ e^{4 - 7x} = 20 - 11 \] \[ e^{4 - 7x} = 9 \] 2. Now take the natural logarithm (ln) of both sides to solve for \( 4 - 7x \): \[ \ln(e^{4 - 7x}) = \ln(9) \] Since the natural logarithm and the exponential function are inverse functions, \( \ln(e^{y}) = y \), you get: \[ 4 - 7x = \ln(9) \] 3. Finally, solve for x: \[ 7x = 4 - \ln(9) \] \[ x = \frac{4 - \ln(9)}{7} \] This is the solution to the equation.
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