Graphical Interpretation of the Derivative
Imagine a skateboard rolling down the curved ramp of a skatepark in Guadalajara. At every instant the skateboard points in one direction: the direction of the line that "touches" the ramp at that point. Can you describe that direction with a single number? Yes: it is the derivative. In this section we connect it with the picture.
The derivative as the slope of the tangent
The derivative $f'(a)$ is the slope of the line that best fits the graph of $f$ at the point with $x$-coordinate $a$. That line is 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 $x = 3$.
Intersection points of two curves
Some problems first ask you to find where two graphs cross and then work with the tangent at one of those points. At an intersection point the two functions have the same value, so you solve $f(x)=g(x)$.
With the chosen intersection point, the tangent is computed as usual: you differentiate $f$, evaluate at that $x$-coordinate, and use the point-slope form.
Given the polynomial functions $f(x) = x^2 - 3x + 2$ and $g(x) = -x^2 + 7x - 6$, first find the intersection points of their graphs. Next, calculate the equation of the…
Given the polynomial functions $f(x) = x^2 - 3x + 2$ and $g(x) = -x^2 + 7x - 6$, first find the intersection points of their graphs. Next, calculate the equation of the tangent line to the function $f(x)$ at the intersection point with the largest x-coordinate.
Derivatives you will need
| Function | Derivative |
|---|---|
| $x^n$ | $n\,x^{n-1}$ |
| sum or difference of terms | differentiate term by term |
| $\sqrt{x}$ | $\dfrac{1}{2\sqrt{x}}$ |
| $e^{kx}$ | $k\,e^{kx}$ |
| $\ln x$ | $\dfrac{1}{x}$ |
| $\dfrac{u}{v}$ | $\dfrac{u'v-uv'}{v^2}$ |
Why believe them? A few quick ideas:
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$\sqrt{x}=x^{1/2}$, and with $n=\tfrac12$ the rule for $x^n$ gives $\tfrac12 x^{-1/2}=\dfrac{1}{2\sqrt{x}}$.
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$e^{kx}$ grows at a rate proportional to its own value, and the constant of proportionality is $k$.
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$\ln x$ is the inverse of $e^x$. Their graphs are symmetric about $y=x$, so the slopes of the tangents at symmetric points are reciprocals. The point $(y, e^y)$ on the graph of $e^x$ has slope $e^y$; its symmetric point $(e^y, y)$ on the graph of $\ln x$ has slope $\dfrac{1}{e^y}$. Since the $x$-coordinate of that point is $e^y$, the slope of $\ln x$ at $x$-coordinate $x$ is $\dfrac1x$.
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Quotient: check it with $\dfrac{x^2}{x}=x$, whose derivative is $1$. The rule gives $\dfrac{2x\cdot x-x^2\cdot 1}{x^2}=1$. ✓
Estimating with the tangent
Near $x=a$, the tangent almost coincides with the curve. That is why it is used to estimate values: $f(x)\approx f(a)+f'(a)(x-a)$.
Find the equation of the tangent line to the graph of the rational function $f(x) = \frac{2x+1}{x-3}$ at the exact point where the graph intersects the y-axis.
Find the equation of the tangent line to the graph of the rational function $f(x) = \frac{2x+1}{x-3}$ at the exact point where the graph intersects the y-axis.
Determine at which exact point on the graph of the radical function $f(x) = \sqrt{x+4}$ the tangent line has a gradient equal to $m = \frac{1}{4}$.
Determine at which exact point on the graph of the radical function $f(x) = \sqrt{x+4}$ the tangent line has a gradient equal to $m = \frac{1}{4}$.
The number of subscribers to a science channel on a video platform grows according to the exponential function $S(t) = 500 e^{0.2t}$, where $t$ is measured in weeks.…
The number of subscribers to a science channel on a video platform grows according to the exponential function $S(t) = 500 e^{0.2t}$, where $t$ is measured in weeks. Find the equation of the tangent line to the curve at time $t = 5$ and use it to estimate linearly the number of subscribers at $t = 6$.
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).
Calculate the exact area of the triangle formed by the x-axis, the tangent line and the normal line to the logarithmic curve $f(x) = \ln(x)$ at the point with…
Calculate the exact area of the triangle formed by the x-axis, the tangent line and the normal line to the logarithmic curve $f(x) = \ln(x)$ at the point with x-coordinate $x = e$.
Horizontal tangent and the sign of $f'(x)$
When the figure shows a parabola, its vertex has a horizontal tangent: there $f'=0$. Since $f'$ of a quadratic is a line, it is enough to know its value at two points (the vertex, where it equals 0, and the slope at another point) to write it.
Find the coordinates of the points on the curve $f(x) = \frac{x^3}{3} - 2x^2 + 3x$ where the tangent line is horizontal.
Find the coordinates of the points on the curve $f(x) = \frac{x^3}{3} - 2x^2 + 3x$ where the tangent line is horizontal.
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.
The figure shows the graph of a quadratic function $f(x)$. From its visible geometrical elements (vertex and axes intercepts), deduce the analytical expression of its…
The figure shows the graph of a quadratic function $f(x)$. From its visible geometrical elements (vertex and axes intercepts), deduce the analytical expression of its derivative function $f'(x)$.
From the graph of $f'$ to the graph of $f$
Sometimes you are given the graph of $f'(x)$, not of $f(x)$, and you are asked to deduce where $f$ increases, where it decreases, and where it has extrema. What matters is whether the curve of $f'$ is above or below the x-axis.
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.
Tangents with a given slope
Sometimes the problem does not give the point, but a condition on the slope. It almost always comes down to solving $f'(x)=m$.
The trajectory of an inspection drone in a vertical plane follows the curve defined by $y = x^3 - 6x^2 + 9x + 2$, where $x$ and $y$ are measured in meters. Determine at…
The trajectory of an inspection drone in a vertical plane follows the curve defined by $y = x^3 - 6x^2 + 9x + 2$, where $x$ and $y$ are measured in meters. Determine at which points along its path the direction of its movement is parallel to the line with equation $y = -3x + 5$.
Determine the coordinates of the points on the curve defined by $f(x) = \frac{x^3}{3} - \frac{x^2}{2} - x + 1$ where its tangent line forms an exact angle of $45^\circ$…
Determine the coordinates of the points on the curve defined by $f(x) = \frac{x^3}{3} - \frac{x^2}{2} - x + 1$ where its tangent line forms an exact angle of $45^\circ$ with the positive x-axis.
Prove algebraically that the graph of the rational function $f(x) = \frac{x^2 + x + 1}{x-1}$ has no tangent line anywhere in its domain that is parallel to the line with…
Prove algebraically that the graph of the rational function $f(x) = \frac{x^2 + x + 1}{x-1}$ has no tangent line anywhere in its domain that is parallel to the line with equation $y = 2x$.
Determine for which set of values of the parameter $k$ the rational function $f(x) = \frac{x^2 + kx + 1}{x-2}$ has no horizontal tangent lines at any point in its domain.
Determine for which set of values of the parameter $k$ the rational function $f(x) = \frac{x^2 + kx + 1}{x-2}$ has no horizontal tangent lines at any point in its domain.
Finding parameters with tangents
When the function has letters ($a$, $b$, $c$…), each geometric fact gives an equation:
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"Passes through $(x_0, y_0)$" $\Rightarrow f(x_0)=y_0$.
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"Tangent of slope $m$ at $x_0$" $\Rightarrow f'(x_0)=m$.
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"Local extremum at $x_0$" $\Rightarrow f'(x_0)=0$.
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"Common tangent of two curves at $x_0$" $\Rightarrow f(x_0)=g(x_0)$ and $f'(x_0)=g'(x_0)$.
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"Continuous and differentiable when joining two pieces" $\Rightarrow$ same value and same derivative at the junction point.
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"Inflection point at $x_0$" $\Rightarrow f''(x_0)=0$ (the derivative of $f'$ is zero).
Why? A fact about position becomes a condition on $f$ and a fact about direction becomes a condition on $f'$. With as many equations as parameters, you get a system that can be solved.
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 $c$.
In the design of a mountain road, a straight section given by the equation $y = 2x$ (for $x \le 1$) must join smoothly with another straight section with equation…
In the design of a mountain road, a straight section given by the equation $y = 2x$ (for $x \le 1$) must join smoothly with another straight section with equation $y = -x + 6$ (for $x \ge 3$) by means of a parabolic transition curve $f(x) = ax^2 + bx + c$ strictly defined on the interval $[1, 3]$. Calculate the values of $a$, $b$ and $c$ so that the road is continuous and differentiable at both junction points ($x=1$ and $x=3$).
Determine the exact values of the parameters $a$, $b$, $c$ and $d$ so that the cubic function $f(x) = ax^3 + bx^2 + cx + d$ has a relative extremum at the point $(0, 2)$…
Determine the exact values of the parameters $a$, $b$, $c$ and $d$ so that the cubic function $f(x) = ax^3 + bx^2 + cx + d$ has a relative extremum at the point $(0, 2)$ and its tangent line at the point $(1, 0)$ is strictly parallel to the line $y = -3x$.
The graphs of the quadratic functions $f(x) = x^2 + ax + b$ and $g(x) = -x^2 + 2x - 1$ are tangent to each other at the point with x-coordinate $x = 1$ (that is, they…
The graphs of the quadratic functions $f(x) = x^2 + ax + b$ and $g(x) = -x^2 + 2x - 1$ are tangent to each other at the point with x-coordinate $x = 1$ (that is, they share the same point and the same tangent line at that point). Calculate the values of $a$ and $b$, and state the equation of this common tangent line.
Let $f(x) = x^3 - 3x^2 + kx$ be the function. Determine the value of the parameter $k$, given that the tangent line to the graph of $f(x)$ at its point of inflection is…
Let $f(x) = x^3 - 3x^2 + kx$ be the function. Determine the value of the parameter $k$, given that the tangent line to the graph of $f(x)$ at its point of inflection is parallel to the angle bisector of the second and fourth quadrants.
Tangents from an external point
If the given point is not on the curve, you cannot compute $f'$ there. You set up the tangent at a generic point $(a, f(a))$ and require it to pass through the given point.
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.
Summary
Go back to the beginning: the direction of the skateboard at a point on the ramp is that of the tangent, and its slope is the derivative at that point.