Precalculus and Algebra

• Function Notation
• Function Transformations
• General Exponential Functions
• Compund Interest

Trigonometry

• Unit Circle

Calculus I

• Using Secant Lines to Approximate the Tangent
• The graphical meaning of \(f'(a)\)
• Linear Approximations and the Tangent Line

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 \(f(x)\)?

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 \(a < 1\)?

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 \(a < 1\)?

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?

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).

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.

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?

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?