Tangent Line to a Curve
Ximena is riding her bike along a curved track and suddenly lets go of the handlebars at one point: the bike shoots off in a straight line, exactly in the direction it was heading. That line is the tangent to the path. With the derivative we can write its exact equation and, along the way, use it to estimate values without a calculator. By the end of this section you'll see how.
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 $x = 2$.
Find the equation of the tangent line to the curve $f(x) = x^2 + 1$ at the point with x-coordinate $x = 1$, as shown in the figure, writing it in the form…
Find the equation of the tangent line to the curve $f(x) = x^2 + 1$ at the point with x-coordinate $x = 1$, as shown in the figure, writing it in the form $y - f(1) = f'(1) \times (x - 1)$.
Determine the equation of the tangent line to the function $f(x) = x^3 - 2x$ at the point with x-coordinate $x = 1$.
Determine the equation of the tangent line to the function $f(x) = x^3 - 2x$ at the point with x-coordinate $x = 1$.
Slope as a Rate of Change
The slope of the tangent is not just geometric information: it tells you how fast $y$ changes with respect to $x$ right at that point. It is the instantaneous rate of change, and its units are the units of $y$ divided by the units of $x$.
If $s(t)$ is the position of a moving object (in meters) at time $t$ (in seconds), then $s'(a)$ is the instantaneous velocity at $t=a$, measured in m/s. On the graph of $s(t)$, the slope of the tangent at $t=a$ is that velocity.
A cyclist training in a velodrome travels $s(t) = 2t^2 + 3t$ meters 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 $s(t) = 2t^2 + 3t$ meters 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 slope of this tangent line represent? State its value and its units.
Tangents to Other Functions
The procedure is always the same: calculate $f(a)$, calculate $f'(a)$, and substitute into $y-f(a)=f'(a)(x-a)$. Only the way you take the derivative changes.
Look at the figure: near $x=1$ the curve and its tangent almost coincide. That is why $\ln x \approx x-1$ for $x$ very close to $1$.
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 $1$?
Horizontal Tangent or 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.
Consider the function $f(x) = x^2 - 2x + 3$. a) Find the equation of the tangent line at the point with x-coordinate $x = 3$. b) Determine the points of intersection of…
Consider the function $f(x) = x^2 - 2x + 3$.
- a) Find the equation of the tangent line at the point with x-coordinate $x = 3$.
- b) Determine the points of intersection of this tangent line with the coordinate axes.
- c) Calculate the area of the triangle formed by the tangent line and the two axes.
Does the Tangent Touch the Curve Only Once?
Tangent does not mean "touches at only one point." That idea comes from the circle, but on other curves the tangent can cross the graph somewhere else.
State, giving reasons, whether the following statement is true or false: "The tangent line to a curve at a point cannot intersect the curve again at another point." If…
State, giving reasons, whether the following statement is true or false: "The tangent line to a curve at a point cannot intersect the curve again at another point." If it is false, give a counterexample using the function $f(x) = x^3$ at the point with x-coordinate $x = 1$.
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
If the given point is not on the curve, we can't use $(a, f(a))$ with its coordinates. Instead, we set up the tangent at a general point $x=a$ and require it to pass through the given point. This produces an equation in $a$.
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.
Determine the point on the curve $f(x) = \ln x$ at which the tangent line passes through the origin, and write down the equation of this tangent. (Hint: formulate the…
Determine the point on the curve $f(x) = \ln x$ at which the tangent line passes through the origin, and write down the equation of this tangent. (Hint: formulate the tangent line at a general point with x-coordinate $x = a$ and impose the condition that it passes through $(0, 0)$.)
Angle Between Curves and Estimates Using the Tangent
Near the point of tangency, the tangent line almost coincides with the curve. Two applications come from this.
Angle between two curves. It is the angle between their tangents at the point where they intersect. With slopes $m_1$ and $m_2$:
$$\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$$Estimating values. For $x$ near $a$: $f(x)\approx f(a)+f'(a)(x-a)$.
Back to the beginning: Ximena's bike leaves along the tangent, and that line is the best approximation of the path near the point where she lets go of the handlebars.
The parabolas $f(x) = x^2$ and $g(x) = 2 - x^2$ intersect forming an angle, which is the angle formed by their tangent lines at the point of intersection, as shown in…
The parabolas $f(x) = x^2$ and $g(x) = 2 - x^2$ intersect forming an angle, which is the angle formed by their tangent lines at the point of intersection, as shown in the figure.
- a) Verify that the two curves intersect at the points with x-coordinates $x = -1$ and $x = 1$.
- b) Calculate the gradients of the tangents to both curves at the point with x-coordinate $x = 1$.
- c) Using the formula for the angle between two lines, $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 \times m_2} \right|$, find the angle formed by the two curves at that point.
It is required to estimate the value of $\sqrt{102}$ without a calculator. a) Find the equation of the tangent line to $f(x) = \sqrt{x}$ at the point with x-coordinate…
It is required to estimate the value of $\sqrt{102}$ without a calculator.
- a) Find the equation of the tangent line to $f(x) = \sqrt{x}$ at the point with x-coordinate $x = 100$.
- b) Use this line to give an estimate of $\sqrt{102}$ to two decimal places.
- c) Given that the actual value rounded to four decimal places is $10.0995$, comment on whether the tangent approximation is an overestimate or an underestimate and explain why.