Graphical interpretation of the derivative
If you know $f'(a)$, you already know a great deal about the graph of $f$ near $x=a$: its exact steepness, whether it rises or falls, and you can draw the line that best approximates it at that point. This section links the number $f'(a)$ with the drawing.
The derivative as the gradient of the tangent
Calculate the equation of the tangent line to the graph of the function $f(x) = x^2 - 4x + 5$ at the point with x-coordinate $x = 3$.
Calculate the equation of the tangent line to the graph of the function $f(x) = x^2 - 4x + 5$ at the point with x-coordinate $x = 3$.
The normal line: perpendicular to the tangent
Determine the equation of the normal line to the curve $f(x) = x^3 - 2x$ at the point with x-coordinate $x = -1$.
Determine the equation of the normal line to the curve $f(x) = x^3 - 2x$ at the point with x-coordinate $x = -1$.
A light ray strikes a parabolic mirror whose cross-section is given by the function $f(x) = \frac{1}{4}x^2$. If the ray hits at the point with x-coordinate $x = 2$, find…
A light ray strikes a parabolic mirror whose cross-section is given by the function $f(x) = \frac{1}{4}x^2$. If the ray hits at the point with x-coordinate $x = 2$, find the equation of the normal line to the surface at that point (which is fundamental in geometrical optics to calculate the angle of reflection).
Reading the sign of $f'(x)$ from a graph
Look at the graph of the function $f(x)$ and the tangent line drawn at the point with x-coordinate $x = 1$. From the coordinates of the points $A$ and $B$ shown on the…
Look at the graph of the function $f(x)$ and the tangent line drawn at the point with x-coordinate $x = 1$. From the coordinates of the points $A$ and $B$ shown on the line, determine the value of the derivative $f'(1)$.
Look at the graph of the polynomial function $f(x)$. Without calculating its analytical expression, determine the sign of the derivative $f'(x)$ at the points with…
Look at the graph of the polynomial function $f(x)$. Without calculating its analytical expression, determine the sign of the derivative $f'(x)$ at the points with x-coordinates $x = -2$, $x = 0$ and $x = 2$, stating whether the function is increasing or decreasing at those points.
From the graph of $f'$ to the graph of $f$
A common type of exercise gives you the graph of $f'(x)$, not that of $f(x)$, and asks you to work out what $f$ is like: where it increases, where it decreases and where it has extrema.
The figure shows the graph of the derivative function $f'(x)$ of a polynomial function $f(x)$. From it, determine analytically the intervals where the original function…
The figure shows the graph of the derivative function $f'(x)$ of a polynomial function $f(x)$. From it, determine analytically the intervals where the original function $f(x)$ is increasing and decreasing, and state the x-coordinates at which $f(x)$ has its relative extrema.
The figure represents the graph of the derivative function $f'(x)$ (composed of two line segments) of a continuous function $f(x)$. Based exclusively on this graph,…
The figure represents the graph of the derivative function $f'(x)$ (composed of two line segments) of a continuous function $f(x)$. Based exclusively on this graph, identify the x-coordinates of the relative extrema of $f(x)$, classifying them as maximums or minimums, and state the x-coordinate of the point of inflection of the original function.
You can do it on paper and send a photo with your answer.
Tangent lines with conditions
Sometimes you are not asked directly for $f'(a)$, but to find tangents that meet a condition: being parallel to a given line, passing through a point that is not on the curve, or adjusting parameters so that the graph passes through certain points with a specific gradient.
An architect is designing the main arch of a bridge that takes the shape of a parabola with equation $f(x) = ax^2 + bx + c$. It is known that the arch passes through the…
An architect is designing the main arch of a bridge that takes the shape of a parabola with equation $f(x) = ax^2 + bx + c$. It is known that the arch passes through the point $(0, 5)$ and that at the point $(2, 9)$ the tangent line has the equation $y = 4x + 1$. Calculate the exact values of the parameters $a$, $b$ and $c$.
Find the equations of all the tangent lines to the curve $f(x) = x^2 - 2x + 3$ that pass through the origin $(0,0)$. Note: keep in mind that the origin does not lie on…
Find the equations of all the tangent lines to the curve $f(x) = x^2 - 2x + 3$ that pass through the origin $(0,0)$. Note: keep in mind that the origin does not lie on the curve.