Calculus - Basic functions 1

Q1: Draw the graphs

Determine the function equation of each function described below, and sketch the graph using a table of values. Also, solve an equation to find a value for $x$ with $f(x)=0.5$.

1. The constant function with height $y=2$.
2. The linear function with slope $2$ and $y$-intercept $-1$.
3. The exponential function with base $e$.
4. The logarithmic function with base $e$.
5. The polynomial with the $x$-intercepts $-1, 0$ and $2$, passing through the point $P(4|3)$. Hint: Use the calculator (poly-solve) to find $x$ with $f(x)=0.5$.
6. The different types of power functions passing through the point $P(1|1)$.
7. The three trigonometric functions. Also, explain how they are defined using the unit circle.

Q2: Exponential and linear growth and decay

Find the tree height at day $30$ and the time when the tree has height $100m$.

1. Starting at day $2$ and tree height $0.4m$, a tree grows every third day by the factor $1.1$.
2. Starting at day $1$ and tree height $0.5m$, a tree grows every second day by $2\%$.
3. Starting at day $2$ and tree height $0.4m$, a tree grows every third day by $0.2m$.

Q3: Exponential or linear growth?

A radioactive material decays every 2 years by $1\%$.

1. By how many percent does the material decay every 4 years? And every year?
2. What is the half-life of the material (that is, when is only half of the material left)?

Q4: Find the function equation

1. Find the linear function the passes through the points $A(2|5)$ and $B(4|1)$.
2. Find the exponential function that passes through the points $A(2|5)$ and $B(4|1)$.
3. Find the sine function with amplitude $2$ and period $3$.
4. Find the polynomial of smallest degree that touches the $x$-axis at $x=2$, and passes through the origin and the point $P(3|5)$.

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