# Practice Exam 1b (not done)

## 1. Evaluate the following limits (if they exist) for $g(x)=\dfrac{3x^2-7x+4}{x^2-4}$.

### (a) $\displaystyle \lim_{x \to 0} g(x)$

### (b) $\displaystyle \lim_{x \to 2} g(x)$

### ( c) $\displaystyle \lim_{x \to -2} g(x)$

### (d) $\displaystyle \lim_{x \to 1} g(x)$

## 2. Understanding Derivatives

### (a) Explain the geometric interpretation of a derivative.

### (b) How does the derivative help in understanding the behavior of functions?

### ( c) Calculate the derivative of $h(x)=3x^2$ using the definition of a derivative.

## 3. Given the function $h(x)$:

$$h(x)=\begin{cases} \dfrac{x^2-9}{x-3} & x>3 \\ 2x & x\leq 3 \end{cases}$$

### (a) Compute $\displaystyle \lim_{x \to 3^-} h(x)$

### (b) Compute $\displaystyle \lim_{x \to 3^+} h(x)$

### ( c) Find $h(3)$

### (d) Discuss the continuity of $h(x)$ at $x=3$.

## 4. Determine the equation of the tangent line to the curve $f(x)=x^3-6x^2+11x-5$ at $x=2$.

## 5. Define what it means for a function to be continuous at a point.

## 6. A company determines that it costs $1500 to produce 50 units of a product and $3500 to produce 150 units. If $C(x)$ represents the cost to produce $x$ units, what is the slope of $C(x)$, and what does it signify in this context?

## 7. Simplify the expression $\dfrac{\sqrt{x^3} \cdot x^{1/2}}{\sqrt[4]{x}}$.

## 8. Consider the function $g(x)=2x^3-6x$.

### (a) Calculate the average rate of change of $g(x)$ over the interval $[1,3]$.

### (b) Find $g'(x)$.

### (c) Determine the instantaneous rate of change of $g(x)$ at $x=1$.

### (d) Calculate the equation of the tangent line to $g(x)$ at $x=1$.

## 9. If a business's revenue $R(q)$ from selling $q$ units of a product is given by $R(q) = 100e^{-0.1q}$, where $q$ is in hundreds and $R$ in dollars, interpret the meaning of $R'(50) = -5$.

## 10. Derive the following functions without using the limit definition:

### (a) $f(x)=5x^5-\dfrac{3}{x}+2\sqrt{x}$

### (b) $L(x)=(2x-3)^2$

### (c) $g(x)=\dfrac{x^3-4x+5}{x^{1/2}}$

## 11. Given the graph of a function $f(x)$ (to be provided), answer the following:

### (a) $\displaystyle \lim_{x \to 2^-} f(x)$

### (b) $\displaystyle \lim_{x \to 2^+} f(x)$

### ( c) $\displaystyle \lim_{x \to 0} f(x)$

### (d) $\displaystyle \lim_{x \to 3^-} f(x)$

### (e) $\displaystyle \lim_{x \to 3^+} f(x)$

### (f) $\displaystyle \lim_{x \to 3} f(x)$

### (g) Identify $x$-values where $f(x)$ is not continuous.