Mathematics Test Review: Exponential and Logarithmic Equations

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What is the limit of $x^2 \ln x$ as $x$ approaches 0 from the positive side?

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Find the derivative of $f(x) = \log_5(x^3 + 9x^2 - 2)$.

$f'(x) = \frac{3x^2 + 18x}{(x^3 + 9x^2 - 2) \ln 5}$

Find the derivative of $g(x) = e^{\sqrt{\cot x}}$.

$g'(x) = -\frac{e^{\sqrt{\cot x}}}{2\sqrt{\cot x}} \csc^2 x$

Find the derivative of $h(x) = \arctan(x^2 \cos x)$.

$h'(x) = \frac{2x\cos x - x^2 \sin x}{1 + (x^2 \cos x)^2}$ Signup and view all the answers

Find the derivative of $h(x) = 5x \sec x$.

$h'(x) = 5\sec x + 5x \sec x \tan x$ Signup and view all the answers

Use logarithmic differentiation to find the derivative of $y = (3x+1)^5 e^{\sqrt{2x^5+1}}$.

$\frac{dy}{dx} = \frac{15(3x+1)^4}{3x+1} + \frac{\sqrt{2}(2x^5+1)^{-1/2}(10x^4)e^{\sqrt{2x^5+1}}}{\sqrt{2x^5+1}}$ Signup and view all the answers

What is the derivative of the function $k(u) = \frac{3u+2}{(3u+2)^2}$?

$k'(u) = \frac{-3(3u+2)}{(3u+2)^3}$ Signup and view all the answers

What is the derivative of the function $m(x) = \ln(4x^3 - 2)$?

$m'(x) = \frac{12x^2}{4x^3 - 2}$ Signup and view all the answers

What is the derivative of the function $n(x) = 3x \sin^{-1}(4x^2)$?

$n'(x) = 3\sin^{-1}(4x^2) + 24x^2\csc(4x^2)$ Signup and view all the answers

Find the limit $\lim_{x\to 0} \frac{3x\sqrt{1 - (4x^2)^2}}{24x\sqrt{1 - (4x^2)^2}}$.

The limit is $\frac{1}{8}$. Signup and view all the answers

Find the limit $\lim_{x\to \infty} \frac{7xe^x - 7x}{x}$.

The limit is $7$. Signup and view all the answers

What is the derivative of the function $k(u) = \frac{3u+2}{\sqrt{u^2 - 1}}$?

$k'(u) = \frac{-3(3u+2)}{(u^2 - 1)^{3/2}}$ Signup and view all the answers

What is the inverse function of $k(x) = \frac{3x+2}{x-4}$?

The inverse function of $k(x) = \frac{3x+2}{x-4}$ is $k^{-1}(x) = \frac{2x-3}{4x+2}$. Signup and view all the answers

If $f(x) = x^3 + 9x + 5$, find $(f^{-1})'(5)$.

$(f^{-1})'(5) = \frac{1}{(f^{-1})'(f(5))} = \frac{1}{(f'(f^{-1}(5)))^{-1}} = \frac{1}{(3(f^{-1}(5))^2 + 9)^{-1}}$. Signup and view all the answers

Rewrite $\log_3(z^4 t^8) = 3x^2 y^5$ in terms of multiple logarithms of single variables.

The correct rewriting is: $2\log_3 x + 5\log_3 y - 4\log_3 z - 8\log_3 t = 3x^2 y^5$. Signup and view all the answers

Rewrite $2\ln(4) - 4\ln(x) + \ln(y) - 4\ln(z) + 6\ln(w) + 3\ln(t) = 2$ as a single logarithm.

The correct rewriting is: $\ln\left(\frac{16\sqrt{y}w^6t^3}{x^4z^4}\right) = 2$. Signup and view all the answers

Explain the steps to find the inverse function of $k(x) = \frac{3x+2}{x-4}$.

To find the inverse function of $k(x) = \frac{3x+2}{x-4}$, we need to:1. Solve the equation $y = \frac{3x+2}{x-4}$ for $x$ in terms of $y$. 2. Swap the $x$ and $y$ variables to get the inverse function $k^{-1}(x) = \frac{2x-3}{4x+2}$. Signup and view all the answers

Explain the significance of finding the derivative of the inverse function, $(f^{-1})'(5)$, in the context of the given problem.

Finding $(f^{-1})'(5)$ is important because it allows us to determine the rate of change of the inverse function at the specific point $x=5$. This information can be useful in analyzing the behavior of the original function $f(x)$ and its inverse $f^{-1}(x)$, which is relevant in many applications of inverse functions. Signup and view all the answers

What is the value of x that satisfies the equation $500e^{3x+2} = 15000$?

The value of x that satisfies the equation $500e^{3x+2} = 15000$ is $\frac{\ln(30)}{3} - \frac{2}{3}$. Signup and view all the answers

For what values of x does the equation $\log_2 x + \log_2 (x - 2) = 3$ hold true?

The equation $\log_2 x + \log_2 (x - 2) = 3$ holds true for $x = -2$ and $x = 4$. Signup and view all the answers

Solve the equation $3\ln(2x - 7) = 9$ for x.

The solution to the equation $3\ln(2x - 7) = 9$ is $x = \frac{e^3 + 7}{2}$. Signup and view all the answers

If $75,000 is invested at 6% interest compounded quarterly for 4 years, what is the amount in the account at the end of the investment period? Round your answer to 2 decimal places.

If $75,000 is invested at 6% interest compounded quarterly for 4 years, the amount in the account at the end of the investment period is $92,723.04 (rounded to 2 decimal places). Signup and view all the answers

How long does it take for an investment to double in value if it is invested at 10% interest compounded continuously?

If an investment is made at 10% interest compounded continuously, it takes approximately 69.66 years for the investment to double in value. Signup and view all the answers

Find the equation of the tangent line to the curve $f(x) = 3e^{1-x} \ln(x)$ at the point x = 1, in the form y = mx + b. Give the values of m and b.

The equation of the tangent line to $f(x) = 3e^{1-x} \ln(x)$ at x = 1 is y = 0x + 0, where m = 0 and b = 0. Signup and view all the answers