Q: Find all $$x in$$ $$[0,2\pi]$$ such that $$\tan^2$ x - 3 = 0.$$

$$x = \tan$$^{-1}$$(\sqrt{3}), \tan$$^{-1}$$(-\sqrt{3}) + \pi$$

Q: Find the horizontal asymptote(s) of $$y = \frac{x^3$ + x^2$}{3x^2$ + \sqrt{25x^6$ + x^2$}}.$$

$$y = \frac{1}{8}$$ and $$y = -\frac{1}{8} (as$$ $$x \to \infty$$ and $$x \to -\infty)$$

Question 1

Solve for $$x$$: $$\ln x + \ln(x-3) = \ln 4.$$

Accepted Answer

$$x = 4 or$$ $$x = -1 ($$but only $$x=4 is $$valid)$

Question 2

Find the exact value: $$\sin$$^{-1}$$\left(\sin\left(\frac{8\pi}{7}\right)\right).$$

Accepted Answer

$$\frac{8\pi}{7} - \pi$$

Question 3

Which of the following intervals is guaranteed to contain a root of the equation $$x^3 - 3x = 5$ by the Intermediate Value Theorem?

Accepted Answer

$$(2,3)$$

Question 4

Let $$f(x) = \ln(\frac{1}{3}e$$^{2x}$$ - 3). $$What is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$(\ln 3, \infty)$$

Question 5

Evaluate $$\lim\limits_{x 	o 5^-} \frac{x^2$ - 3x - 10}{|x-5|}.$$

Accepted Answer

$$7$$

Question 6

Suppose $$f(x) is $$continuous on $$[1,6]$$ and $$f(1) = 1.5$$, $$f(2) = 2.2$$, $$f(3) = 2.9$$, $$f(4) = 3.5$$, $$f(5) = 5.5$$, $$f(6) = 5.9. $$How many solutions does $$f(x) = x$$ have on $$[1,6] by $$the Intermediate Value Theorem?

Accepted Answer

2

Question 7

Find all $$x in$$ $$[0,2\pi]$$ such that $$	an^2$ x - 3 = 0.$$

Accepted Answer

$$x = 	an$$^{-1}$$(\sqrt{3}), 	an$$^{-1}$$(-\sqrt{3}) + \pi$$

Question 8

Express $$\cos(\sin$$^{-1}$$(\frac{x}{5})) as an $$algebraic function of $$x$$ for $$|x| \leq 5.$$

Accepted Answer

$$\sqrt{1 - \frac{x^2$}{25}}$$

Question 9

Let $$f(x) = x + 5$$ for $$x \u003e 2$$, and $$f(x) = x - 2$$ for $$x  7$$?

Accepted Answer

$$f^{-1}(x) = x - 5$ for x$$ \u003e 7$$

Question 10

Compute $$\lim\limits_{x 	o 0} \frac{\sin(5x)}{2x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 11

Suppose $$f(x) is $$differentiable and $$f(2) = 3$$, $$f'(2) = 5. $$Compute $$\lim\limits_{h 	o 0} \frac{f(2 + 4h) - f(2)}{3h}.$$

Accepted Answer

$$\frac{20}{3}$$

Question 12

Find the horizontal asymptote(s) of $$y = \frac{x^3$ + x^2$}{3x^2$ + \sqrt{25x^6$ + x^2$}}.$$

Accepted Answer

$$y = \frac{1}{8}$$ and $$y = -\frac{1}{8} (as$$ $$x 	o \infty$$ and $$x 	o -\infty)$$

Question 13

Let $$f(x) = x^2$ - ax - b$$ for $$x \u003c 1$$ and $$f(x) = \sqrt{ax + b}$$ for $$x \geq 1. $$For which values of $$a$$ and $$b is$$ $$f(x)$$ continuous at $$x=1$$?

Accepted Answer

$$a = 2, b = -1$$

Question 14

Suppose $$f(x) = x \sin\left(\frac{1}{x}ight)$$ for $$x 
eq 0$$, $$f(0) = 0. Is$$ $$f(x)$$ continuous and differentiable at $$x=0$$?

Accepted Answer

Continuous but not differentiable at $$x=0$$

Question 15

Let $$f(x) = x \cos(x). $$Find the $$y-$$intercept of the tangent line to $$f(x) at$$ $$x = \frac{\pi}{2}.$$

Accepted Answer

$$y = \frac{\pi}{2} - 1$$

Question 16

Suppose $$f(3) = 2$$ and $$f'(3) = -3. $$Compute $$\lim\limits_{x 	o 3} \frac{x^2$ f(x) - 18}{x-3}.$$

Accepted Answer

$$-6$$

Question 17

Suppose $$f is $$invertible and $$f(2) = 3$$, $$f'(2) = 5. $$What is $$\left(\frac{1}{f$$^{-1}$$(x)}\right)' at$$ $$x=3$$?

Accepted Answer

$$-\frac{1}{15}$$

Question 18

Let $$f(x) = x^2$ and L$ be a tangent line to f$ passing through (1$$, -3)$$. What is the equation of L$?

Accepted Answer

$$y = -2x - 1$$

Comprehensive Calculus Exam 1 Study Guide: Step-by-Step Guidance

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