Differential Calculus Pretest

Welcome to the Differential Calculus Pretest. Attempt to complete all of the following questions in under 90 minutes. The solutions can be found by clicking this link: Differential Calculus Pretest Solutions.

# Pretest (90 Minutes):

1. Given the equation of the linear line below, state the following: slope, x-intercept(s) and y-intercept(s).

(1)
$$y - 5 = -4(x - 2)$$

2. Determine the equation of a line that passes through the point (2, 6) and is perpendicular to the line below. Give your answer in the form of y = mx + b.

(2)
$$0 = -2y + 8x -6$$

3. Given the following quadratic equation, determine the x-intercept(s) if there are any, y-intercepts, as well as the vertex. Also determine if the quadratic equation opens up or down. You may need to utilize the quadratic formula which is provided below for reference.

(3)
$$y = 3x^2 -5x + 7$$
(4)
\begin{align} x = -b \pm \frac{\sqrt{b^2 - 4ac}}{2a} \end{align}

4. Determine the roots of the following expression:

(5)
$$4x^2 - 36$$

5. Simplify the following expression and state the equation(s) of any vertical asymptote(s).

(6)
\begin{align} \frac{x^2 - 6x + 9}{x - 3} \end{align}

6. What is the area of the rectangle bounded by the lines x = 3, x = 7, y = 2, and y = -6?

7. State the value of the following without a calculator:

(7)
\begin{align} sin(60^\circ), tan(\frac{5 \pi }{6}), csc(30^\circ) \end{align}

8. For what value(s) of x does sin(x) = cos(x)?

9. Prove the following trigonometric identity:

(8)
\begin{align} tan(x) \frac{1 - sin^2(x)}{(sin(x))^2} = cot(x) \end{align}

10. State the domain of the following functions:

(9)
\begin{align} y = \sqrt{(x + 3)} \end{align}
(10)
$$f(x) = ln(2x - 3)$$
(11)
$$y = 2x - e^x$$

11. Determine the value of x without the use of a calculator.

(12)
$$x = log_{2}(128)$$

12. Determine the value of x without the use of a calculator.

(13)
$$x = e^{2ln(5)}$$