Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
Find the equation of the line passing through \(\left(1,2\right)\) with gradient \(\frac{1}{2}\).
Find the gradient of \(f\left(x\right)=2x^{2}-3x+4\) at the point \(\left(2,6\right)\).
Solutions
Get Ready
Questions
Think back to how we defined the gradient of a curve.Why is the concept of ‘tangent’ so important to this idea?
Solutions
Notes
The equation of a straight line with gradient \(m\) that passes through the point \(\left(x_{1},y_{1}\right)\) can be found using:$$y-y_{1}=m\left(x-x_{1}\right)$$
The tangent to a curve at any point is the straight line that just touches the curve at that point.
The normal to a curve at any point is the straight line that passes through the point and is perpendicular to the tangent.
The gradient of the tangent to \(y=f\left(x\right)\) at \(x=a\) is given by$$m_{T}=f’\left(a\right)$$
The gradient of the normal to \(y=f\left(x\right)\) at \(x=a\) is given by$$m_{N}=-\frac{1}{f’\left(a\right)}$$
Examples and Your Turns
Example
Find the equation of the tangent to the curve \(f\left(x\right)=2x^{2}-3x+4\) at the point \(\left(2,6\right)\).
Example
Find the equation of the normal to the curve \(f\left(x\right)=3x^{2}-8x-2\) at the point where \(x=2\).
Your Turn
Find the equation of the tangent and normal to the curve \(f\left(x\right)=\cos x +e^{x}\) at the point \(x=0\), giving your answer in the form \(ax+by+d=0\).
Your Turn
Find the equation of the tangent and normal to the curve \(g\left(x\right)=x^{3}-5x^{2}-x^{\frac{3}{2}}+22\) at the point \(x=4\), giving your answer in the form \(ax+by+d=0\).
Your Turn
Find the equation of the tangent to \(f\left(x\right)=e^{2x}\cos x\) at the point \(x=0\).
Your Turn
Find the equation of the normal to \(y=\ln \left(x^{2}+1\right)\) at the point \(x=1\).
Your Turn (HL)
Find the equation of the tangent to \(x^{2}+y^{2}=25\) at the point \(\left(3,4\right)\).
Your Turn (HL)
Find the equation of the normal to \(x^{2}y+y^{2}=5\) at the point \(\left(2,1\right)\).
Your Turn
Show that there are two points on the graph \(y=x^{2}\left(x-2\right)\) at which the gradient is equal to 4. Find the equation of the tangents at these points.
Your Turn
The tangent at the point \(P\) on the curve \(y=x^{2}+1\) passes through the origin. Find the possible coordinates of \(P\).
Your Turn
The curve \(C\) has equation$$\left(x+y\right)^{3}=16x^{2}-8y$$
(a) Find an expression for \(\frac{dy}{dx}\)
The point \(P\left(1,1\right)\) lies on \(C\).
(b) Show that the normal to \(C\) at \(P\) has equation \(y=-x+2\).
(c) Find the coordinates of the other point where the normal to \(P\) at \(C\) intersects \(C\).
Your Turn
Consider the curve defined by \(f\left(x\right)=\frac{\sin\left(2x\right)}{e^{x}}\) for \(0\le x \le \pi\).
a) Show that \(f’\left(x\right)=\frac{2\cos\left(2x\right)-\sin\left(2x\right)}{e^{x}\)
b) Find the equation of the normal to the curve at the point \(P\left(\frac{\pi}{2},0\right)\). Give your answer in the form \(ax+by+d=0\).
c) The tangent to the curve at point \(Q\) is horizontal. Find the \(x\)-coordinate of \(Q\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
If a function has a “sharp corner” (like \(y=\left|x\right|\) at \(x=0\)), why can we not find a well defined tangent there?
Explain the process for finding the tangent to a curve at a point.
Common Mistakes / Misconceptions
The most common mistake is forgetting to do the negative reciprocal for the gradient of the normal.
Another common pitfall is to forget to find the \(y\)-coordinate for the given \(x\)-coordinate by substituting into the original function (not the derivative).
Connecting This to Other Skills
You need to be able to find the equation of Straight Lines (2.1) through a point, including with perpendicular gradients.
You need to be able to perform all the different differentiation techniques such as Differentiating Polynomials (4.5), the Chain Rule (4.7), Product Rule (4.8) and Quotient Rule (4.9), as well as Implicit Differentiation (4.10) and Differentiating Exponentials and Logarithms (4.11) and Differentiating Trig (4.12).
This skill will help when finding Stationary Points (4.14)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?