Brilliance Math Tutoring

😅 Today’s simple example comes with a sprinkle of derivations and covers logarithms. At its core, logarithms allow us to graph functions that relate small values to much larger values (and vice versa).

✍🏾 Given a function, can you find its rate of change at a certain value? This one should be easy 💯

💡 Something pretty cool about math is how we can determine a value, t, within a particular interval of a continuous, differentiable function, that represents a tangent line that is parallel to a line connecting the endpoints defined by the interval. Simply put, there is a value, t, that satisfies the following condition.

s’(t) = [s(b) - s(a)] / (b - a)
Where “a” and “b” are the lower and upper ends of the given interval.

In essence, we are working backwards to satisfy arbitrary conditions.

✔ Step 1: Verify continuity. The function exists at every point in the interval.
✔ Also step 1: Verify differentiability. Does the given function even have a derivative?
✔ Step 2: Find the average rate of change.
✔ Step 3: Calculate the derivative of the given function.
✔ Step 4: Apply the today’s theorem to find t.

🎈🤔 Check out this one! A balloon that enlarges at a rate of 6.5 cubic inches per 30 seconds equates to 13 cubic inches per minute. Now, to find the rate increase of the balloon’s radius at a certain volume, we must first determine the instantaneous radius when the balloon’s volume is 12 cubic inches.

Using the initial formula, V = (4/3)•𝜋•(r3), we find the instantaneous radius to be roughly 1.42 inches. Substituting 1.42 for “r” in equation [1], we find the balloon’s radius to be increasing at roughly half an inch per minute.

📝👀 At first glance, we see perfect squares. This will make today’s exercise marginally easier. In step 1, “x - 1” is found to be a factor after some trial-and-error. Trial-and-error meaning substitute “1” for “x” in f(x) and verify whether or not f(x = 1) = 0. If true, “x - 1” is a factor of the given polynomial. If false, “x - 1” is not a factor. Proceed by testing 2, then 3, then 4, and so on. (The same can be done with negative numbers).

Next, we separate “x - 1” into two ‘groups’ with each group being multiplied by a distinct coefficient. The sum of these coefficients is in fact the quotient of “4(x^3) - 4(x^2) - 25x + 25” divided by “x - 1.” Further factorization of the said quotient, gives us two additional factors: “2x + 5” and “2x - 5.” Solving each of the three factors results in the x-values shown in step 1.

Step 2 illustrates a derivation, f’(x), and domains where f(x) increases and decreases.

Step 3 covers an additional derivation, f”(x), and domains where f(x) “opens up” and “opens down.”

💡 Again, extrema are nothing more than maximum and minimum values of a function over a specified range of x-values (i.e., domain). Finding those x-values requires the first-order derivative of the given function. After which we can substitute those x-values into the original equation to find their corresponding y-values. Result? A list of points that satisfies the given interval — in this case (-3, 3). In other words, the lower and upper ends of the domain are exclusively -3 and 3, respectively.

📝 The opposite of last week's discussion topic, derivations, are also fun and involved in its own unique ways. Depending on how the given function presents itself (whether a product of two functions, quotient of two functions, a sum, difference, or a composition), the process can easily become intricate. Today's examples examine a few of these instances.

🤔 Try this! Identify the error(s) in the derivation of v(w) in part a).

📝 Integration is arguably a fun and involved topic. Depending on the task at hand, and understanding the many ways we can integrate, the process can range from simple to complex. Luckily, today's example is simple.

Step 1: Find the anti-derivative of g(x).
Step 2: Substitute the upper and lower limits accordingly.
Step 3: Solve!

As you can imagine, integration is pivotal in fields like physics and engineering. It helps calculate areas, volumes, and even predict future trends.

✍🏾 Try this out: Integrate z(p) from -2 to 4.

🚨 What exactly does it mean to solve an equation with a trigonometric function? In short, it is finding (a) value(s) that satisfies the condition of the given equation, g(x) = 0. This example states there being values of x that make "2•sin(x) - 1" equal to zero.

Now, what does it mean to constrain that list of solutions to a given interval? Simple. After determining all possible solutions of the given equation, values less than 0 (and values larger than & equalling 2𝛑 ) are ignored in our list of final answers.

✍🏾 As practice: Given the equation of k(x), what is the amplitude, period, and vertical shift?

🚨 Given: Scores from five (5) random students.

Task: Approximate the linear function that describes the relation between the points.

Solution: Use Linear Regression to approximate the line that best represents the given data, where "b" is the slope of the line, and "a" is the y-intercept in the following formula.

y = bX + a

🚨 Another example of derivatives, slopes, and concavity. As outlined below, derivatives are needed to find slopes and concavity.

1st order derivative -- the derivative taken once.
2nd order derivative -- the derivative taken a second time.

🚨 Here, two arbitrary functions are composed to form a new function. That new function depends on how we are asked to manipulate the arbitrary functions. When tasked with substituting a variable with a value, and verifying a true or false statement, we get this exercise.

The between A and B is __________.

The , L, of AB is then __________.