Tangent line to a curve
You already know how to differentiate any function. Now it is time to use the derivative for what made it famous: finding the exact equation of the tangent line, the line that best approximates the curve locally at a point. It is the most direct geometric application of the derivative, and the gateway to the maxima, minima and optimisation problems that come next.
The equation of the tangent line
Calculate the gradient of the tangent line to the graph of the function $f(x) = x^2 - 3x$ at the point with x-coordinate $x = 2$.
Calculate the gradient of the tangent line to the graph of the function $f(x) = x^2 - 3x$ at the point with x-coordinate $x = 2$.
Determine the equation of the tangent line to the function $f(x) = x^3 - 2x$ at the point with x-coordinate $x = 1$.
Determine the equation of the tangent line to the function $f(x) = x^3 - 2x$ at the point with x-coordinate $x = 1$.
Tangents to non-polynomial functions
The same procedure —calculate $f(a)$, calculate $f'(a)$, substitute into $y-f(a)=f'(a)(x-a)$— works for any function, as long as you know how to differentiate it.
The figure shows the branch of the hyperbola $f(x) = \frac{1}{x}$ corresponding to $x > 0$ and the point with x-coordinate $x = 2$. Find the equation of the tangent line…
The figure shows the branch of the hyperbola $f(x) = \frac{1}{x}$ corresponding to $x > 0$ and the point with x-coordinate $x = 2$. Find the equation of the tangent line at this point and verify that it intersects the y-axis at $y = 1$.
Find the equation of the tangent line to the curve $f(x) = \ln x$ at the point with x-coordinate $x = 1$. How is this line related to the values of $\ln x$ when $x$ is…
Find the equation of the tangent line to the curve $f(x) = \ln x$ at the point with x-coordinate $x = 1$. How is this line related to the values of $\ln x$ when $x$ is very close to $1$?
Points with a horizontal tangent or a tangent parallel to a line
At which point on the graph of the function $f(x) = x^2 - 4x + 3$ is the tangent line horizontal? State the coordinates of the point.
At which point on the graph of the function $f(x) = x^2 - 4x + 3$ is the tangent line horizontal? State the coordinates of the point.
Find the point on the parabola $f(x) = x^2$ at which the tangent line is parallel to the line $y = 4x - 1$.
Find the point on the parabola $f(x) = x^2$ at which the tangent line is parallel to the line $y = 4x - 1$.
Find the points on the graph of $f(x) = x^3 - 3x + 2$ at which the tangent line is parallel to the line $y = 9x + 5$, and state the coordinates of each one.
Find the points on the graph of $f(x) = x^3 - 3x + 2$ at which the tangent line is parallel to the line $y = 9x + 5$, and state the coordinates of each one.
The normal line
The normal line to a curve at a point is the line perpendicular to the tangent line at that point. Find the equations of the tangent line and the normal line to the…
The normal line to a curve at a point is the line perpendicular to the tangent line at that point. Find the equations of the tangent line and the normal line to the parabola $f(x) = x^2$ at the point with x-coordinate $x = 1$.
Tangents from an external point
When the given point does not lie on the curve, you cannot use $f(a)$ directly: you have to set up the tangent at a general point $x=a$ of the curve and require it to pass through the given external point, which gives an equation in $a$ with (normally) two solutions.
A cyclist training in a velodrome travels $s(t) = 2t^2 + 3t$ metres during the first $t$ seconds. a) Find the equation of the tangent line to the graph of $s(t)$ at time…
A cyclist training in a velodrome travels $s(t) = 2t^2 + 3t$ metres during the first $t$ seconds.
- a) Find the equation of the tangent line to the graph of $s(t)$ at time $t = 2$.
- b) What physical quantity does the gradient of this tangent line represent? State its value and its units.
From the point $P(0, -4)$, outside the parabola $f(x) = x^2$ and shown in the figure, tangent lines to the curve are to be drawn. a) Write down the equation of the…
From the point $P(0, -4)$, outside the parabola $f(x) = x^2$ and shown in the figure, tangent lines to the curve are to be drawn.
- a) Write down the equation of the tangent line to the parabola at a general point with x-coordinate $x = a$.
- b) Apply the condition that it passes through $P$ and find the equations of the two possible tangent lines, stating the points of tangency.