# Sample Exam A

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### **1.**

**a)** Determine the largest possible domain of the following function.
$$
f(x) = \arcsin \frac{2x-3}{7} - \sqrt{4 - 3x}
$$

**b)** Determine the inverse function of the folloing function.
$$
f(x) = e^{2x - 7} + 5
$$

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### **2.**

**a)** Determine the limit of the following sequence.

$$
a_n = \frac{2^{2n+3}-5}{3^{n-1} - 4^{n+2}}
$$

**b)** Compute the following limit.
$$
\lim_{x \to 1^-} \frac{x^2 + 3x - 2}{x^2 + 3x - 4}.
$$

**c)** A function is defined on the domain $D_f = \mathbb{R} \setminus \{-1, 2\}$ with the following properties:

* $\displaystyle\lim_{x \to -\infty} f(x) = 2$
* $\displaystyle\lim_{x \to -1} f(x) = 1$
* $\displaystyle\lim_{x \to 2^-} f(x) = -\infty$
* $\displaystyle\lim_{x \to 2^+} f(x) = \infty$
* $\displaystyle\lim_{x \to \infty} f(x) = \infty$

Draw a function with these properties.

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### **3.**

Find the derivative of the following functions.

**a)**
$$
f(x) = \sqrt{5x} \cdot \cos(3x)
$$

**b)**
$$
g(x) = \frac{x^2 - \ln x}{2^x}
$$

**c)**
$$
h(x) = \sin^4(3 - 2x)
$$

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### **4.**

Examine the function for monotonicity and extreme values.
$$
f(x) = \frac{e^{-2x}}{x^2}
$$

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### **5.**

Find the equation(s) of the tangent line(s) to
$$
f(x) = x^3 - 3x^2 - 11x + 8
$$
that are parallel to the line
$$
2x + y = 1.
$$

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### **6.**

**a)**
$$
\int \frac{4x - \sqrt[3]{x }+ 3}{x^2} \ dx
$$

**b)**
$$
\int_0^1 2^{3x - 1} \ dx
$$

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### **7.**

**a)**
$$
\int (3x + 7) \cdot \cos x\ dx
$$

**b)**
$$
\int \frac{1}{x - (\sqrt[3]{x })^2} \ dx
$$

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### **8.**

**a)** Determine the volume of the solid obtained by rotating the curve of
$$
f(x) = \frac{2}{\sqrt[3]{2x-1}}
$$
around the x-axis over the interval
$$
[1, 14].
$$

**b)** Approximate
$$
\int_0^{\pi / 6} \sin x \ dx
$$
using the simple trapezoidal rule.