Wan Mekwi

Apply the factor and remainder theorems to manipulate polynomial expressions

Write the expression $ax^2+bx+c$ in completed square form $a(x+p)^2+k$.

Calculating limits using algebraic techniques

Evaluate a rational limit using algebra

Evaluate $\displaystyle \lim_{x\to k} \frac{x+a}{\sqrt{x+b} - c}$ using algebra

Evaluate a rational limit using algebra

Evaluate $\displaystyle \lim_{x\to 0} \frac{\sqrt{ax+b} - d}{cx}$ using algebra

Evaluate $\displaystyle \quad \lim_{x\to a} \frac{ex+d}{ax^2+bx+c} \quad $ algebraically.

Evaluate $\displaystyle \quad \lim_{x\to a} \frac{ax^2+bx+c}{ex+d} \quad $ algebraically.

Evaluate $\displaystyle \quad \lim_{x\to a} \frac{ax+b}{x^2+c} \quad $ algebraically.

More work on differentiation with trigonometric functions

Introductory function notions

This assessment reviews some of the material covered in the first lecture session. 

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Given $\rho(t)=\rho_0e^{kt}$, and values for $\rho(t)$ for $t=t_1$ and a value for $\rho_0$, find $k$. (Two examples).

Apply and combine logarithm laws in a given equation to find the value of $x$.

Solve for $x$ each of the following equations: $n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.

Solve for $x$: $\log(ax+b)-\log(cx+d)=s$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Express $\log_a(x^{c}y^{d})$ in terms of $\log_a(x)$ and $\log_a(y)$. Find $q(x)$ such that $\frac{f}{g}\log_a(x)+\log_a(rx+s)-\log_a(x^{1/t})=\log_a(q(x))$

Given a sum of logs, all numbers are integers,

$\log_b(a_1)+\alpha\log_b(a_2)+\beta\log_b(a_3)$ write as $\log_b(a)$ for some fraction $a$.

Also calculate to 3 decimal places $\log_b(a)$. 

Use the rule $\log_a(n^b) = b\log_a(n)$ to rearrange some expressions.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).

Solve exponential equation of the form \[ a^{kx}=b^{kx+m}. \]

Solve exponential equation of the form \[ a^{kx}=a^{m}. \]

This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.

Questions on powers, the laws of indices, and exponential growth.

Long division of a quartic by a quadratic

Quotient and remainder, polynomial division.

Questions involving various techniques for rearranging and solving quadratic expressions and equations