IB Mathematics SL
Topic 7 Notes - Introduction to Differential Calculus
These notes will cover basic differentiation, slopes of curves and equations of tangents.
In Calculus there are different ways to ask you to do the same thing. So this topic seems longer and more
confusing than it actually is because you may be asked to do the SAME thing in several different ways.
* Differentiate the following.
* Find the derivative of the following.
* Find dy/dx read as ‘dee y by dee x’.
* Find f ’(x) read as ‘f dashed x’.
* Find an expression for slope.
* Find an expression for gradient.
Differentiation
Differentiation is the process used to find an expression for slope/gradient.
To find the derivative, complete the two steps below.
Step1: Multiply the exponent by the coefficient to find the new coefficient.
Step 2: Subtract 1 from the exponent to get the new exponent.
We can differentiate an expression with more than one term in the same way.
To differentiate an expression without an exponent, drop the variable from the expression.
When you differentiate a constant number (with no variable), the answer is zero.
Differentiating Fractions
When asked to differentiate a fraction, follow the steps below.
Step 1: Split the terms
Step 2: Simplify the terms
Step 3: Differentiate each term
Step 4: Take the variables with the negative exponents down to make the exponents positive.
Finding the slope of curves when x is given
When asked to find the slope of a curve, first differentiate then plug in the given x value.
Finding y when x is given
When you are given the x coordinate and asked to find the y-coordinate, you can either:
*Plug x into the original equation
OR
*Use the calculator to graph the original equation then press
2nd èTrace [Calc] è Scroll down to value and press [ENTER] ètype the given x value and press [ENTER]
Finding the Equation of the Tangent to a Curve
The tangent to the curve is a line that touches the given curve at one point.
To find the equation of the tangent, first find the slope (by differentiating) then find the y coordinate. Plug in
slope, x and y into y = mx + b and solve for b to find the y-intercept.
Finding a Point with a Given the Slope
If you are asked for the point on a curve with a given slope, follow the steps in the example below.
First differentiate the expression to find an expression for the slope.
Plug into the original equation or use the graphing calculator to find the value of y.
Function Notation
Read below to make sure you understand questions in which function notation is used.
f (x) can be used instead of y.
f (2) is the value of y when x = 2.
f ‘ (x) can be used instead of dy/dx.
f ‘ (2) is the slope when x is 2.
f ‘’ (x) is the 2nd derivative of the function f (x).
Example: If f (x) = 2x3
Then f ‘ (x) = 6x2 ; f ‘’ (x) = 12x
Increasing and Decreasing FUNCTIONS
You may be asked where a function is increasing or decreasing. To answer this question, simply look for where
the graph is going up or down.
Increasing and Decreasing Slope
More thought is required when asked if the slope of the curve is increasing or decreasing.
If asked what is happening to slope, consider where the slope of the graph is positive, negative or zero. Also
consider the steepness of the graph at various points.
If we are given the equation of a function, we can find where the function is increasing and where it is
decreasing. The concepts of increasing and decreasing are closely linked to intervals of a function’s domain. On
an interval where the function is increasing, an increase in x produces an increase in y. On an interval where the
function is decreasing, an increase in x produces a decrease.
We complete the sign diagram by marking positive or negative signs, depending on whether f ‘(x) is
For the functions that we deal with in this course:
•
f (x) is increasing on S if f ‘(x) > 0 for all x in S.
•
f (x) is strictly increasing if f ‘(x) > 0 for all x in S.
•
f (x) is decreasing on S if f (x) ≥ 0 for all x in S.
•
f (x) is strictly decreasing if f ‘(x) < 0 for all x in S
SIGN DIAGRAMS
Sign diagrams for the derivative are extremely useful for determining intervals where a function is increasing or
decreasing.
The critical values for f ‘(x) are the values of x for which f ‘(x) = 0 or f ‘(x) is undefined. When f ’(x) = 0, the
critical values are shown on a number line using tick marks.
When f ‘(x) is undefined, the critical values are shown with a vertical dotted line.
positive or negative, in the intervals between the critical values.
Finding the Coordinates of a Maximum or Minimum Point
It is EXTREMELY important to remember that the derivative is always zero at the maximum and minimum
points.
Calculus Applications
You are expected to be able to apply your knowledge of algebra and calculus to problems which involve
maximum and minimum.
The example below shows how calculus can be applied.