# Interactive Mathematics Graphics

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# Precalculus and Algebra

## Function Notation

Directions: Try moving around the top slider, which adjusts the input $$x$$ which we are using to compute $$f(x)$$.

• What happens to the output as you change the input?
• What is $$f(0)$$?
• For what $$x$$ is $$f(x) = 0$$?
• How many $$x$$ are there with $$f(x) = 5$$? What are the values of these $$x$$?
• What is the approximate range of $$f(x)$$?

## Function Transformations

Changing the base of the exponential changes the graph significantly.

Directions: Move the slider around to examine the graph of $$y=a^x$$ for different values of $$a$$. What happens when $$a > 1$$? What about when $$a = 1$$, or when $$a < 1$$?

## General Exponential Functions

Changing the base of the exponential changes the graph significantly.

Directions: Move the slider around to examine the graph of $$y=a^x$$ for different values of $$a$$. What happens when $$a > 1$$? What about when $$a = 1$$, or when $$a < 1$$?

## Compound Interest

Understanding the impact of different interest rates is an important life skill. Use this graphic to get a concrete picture of the impact of the interest on the principal.

Directions: The top slider controls the interest rate (as a decimal), the middle slider controls the number of compoundings per year, and the bottom slider controls the principle (in thousands).

• Most investments have interest rates between 1% and 5%. What does this look like?
• Most credit cards have interest rates around 25% or 30%. What does this look like?
• How does the graph change when you adjust the number of compoundings per year?
• What impact does changing the principal have?

# Trigonometry

## The Unit Circle

Understanding the impact of different interest rates is an important life skill. Use this graphic to get a concrete picture of the impact of the interest on the principal.

Directions: The slider controls the angle (in radians).

# Calculus I

## Using Secant Lines to Approximate the Tangent

Computing the slope of the line tangent to $$f(x)$$ at $$a$$ is tricky. Our formula for the slope of a line requires two points, but the tangent is defined using a single point.

As $$h$$ gets smaller, the slope of the secant line between $$(a,f(a))$$ and $$(a+h,f(a+h))$$ approaches the slope of the tangent to $$f(x)$$ at $$a$$.
When $$h$$ is very small, the slope of the secant is nearly identical to the slope of the tangent.

Directions: Try moving the slider for $$h$$. What happens when you make $$h$$ closer to 0? Next try moving the slider for $$a$$, to approximate the tangent at a different point.

## The graphical meaning of $$f'(a)$$

There are two sides to the definition of $$f'(x)$$. Graphically, $$f'(a)$$ is the slope of the tangent to $$f(x)$$ at $$a$$. Algebraically, $$\displaystyle f'(a) = \lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$.

Directions: Move the slider to view the tangent line and $$f'(a)$$ for different values of $$a$$. What do you notice?

## Linear Approximations and the Tangent Line

The line tangent to $$f(x)$$ at $$a$$ gives a good approximation to $$f(x)$$ for $$x$$ near $$a$$. This interactive graphic can help you get a sense of how accurate this approximation is.

Directions: Try moving around the top slider, which adjusts the point $$x$$ where we want to approximate $$f$$. What happens to the error and percent error?
Now use the bottom slider to adjust the location $$a$$ of the tangent line approximation to $$f$$. For example, center the linear approximation to $$f$$ at the point $$a= 0$$. Now how does adjusting the point $$x$$ impact the error and percent error?