Tangents & Normals at a given point: notes and practice questions

Consider the function $f(x) = x^{3} + 5x^{2} - x + 5$.

(a) Find $f'(x)$.

(b) Find the equation of the tangent line at $x = 1$.

(c) For the equation of the normal line at $x = 1$.

The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = -2x + 9$ is the tangent to the graph of $f$ at $x = 3$.

(a) Write down the value of $f'(3)$.

(b) Find $f(3)$.

The function $g$ is defined for all $x \in \mathbb{R}$ where $g(x) = x^2 - 1$ and $h(x) = f(g(x) )$.

(c) Find $h(2)$.

(d) Hence, find the equation of the tangent to the graph of $h$ at $x = 2$.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

Consider the functions $f(x) = \ln(x-p)$ and $g(x) = \frac{1}{4}x^2 + q$ where $p, q \in \mathbb{R}$.

(a) Find $g'(x)$.

The graphs of $f$ and $g$ have a common tangent at the point where $x = 2$.

(b) Show that $p = 1$.

(c) Hence, find the value of $q$.

A function $f$ is defined by $f(x) = \frac{4x-1}{2x+3}$, where $x \in \mathbb{R}, x \neq -\frac{3}{2}$.

The graph of $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider the function $g(x) = mx - \frac{1}{3}$, where $m \in \mathbb{R}, m \neq 0$.

(i) Write down the number of solutions to $f(x) = g(x)$ for $m < 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$ for which $f(x) = g(x)$ has two distinct solutions for $x \le 0$.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

The functions $f$ and $g$ are defined by $f(x) = \sin x$ and $g(x) = \cot x$, for $0 < x < \frac{\pi}{2}$.

The curves $y = f(x)$ and $y = g(x)$ intersect at a point P whose x-coordinate is $k$.

Show that $\sin^2 k = \cos k$.

Hence, show that the tangent to the curve $y = f(x)$ at P and the tangent to the curve $y = g(x)$ at P are perpendicular.

Find the value of $\cos k$. Give your answer in the form $\frac{a+\sqrt{b}}{c}$, where $a, c \in \mathbb{Z}$ and $b \in \mathbb{Z}^+$.

Consider the functions $f(x) = k - (x-h)^2$ and $g(x) = \ln(x-1) + 2$ where $h, k \in \mathbb{R}$.

The graphs of $f$ and $g$ have a common tangent at $x=2$.

(a) Find $g'(x)$.

(b) Show that $h = \frac{5}{2}$.

(c) Hence, find the value of $k$.

A company models the average cost per unit, $C(x)$, in thousands of dollars, for producing $x$ thousand units of a specialized component using the function $C(x) = \frac{kx-3}{x-k}$, where $x > 0$ represents the number of units in thousands, and $k$ is a positive constant.

(a) Write down the equations of the vertical and horizontal asymptotes of the graph of $C$.

(b) Show that $C'(x) = \frac{3-k^2}{(x-k)^2}$.

(c) Show that the graph of $C$ has no turning points.

(d) Find the equation, in terms of $k$, of the normal to the graph of $C$ at $x = 1$.

(e) The horizontal and vertical asymptotes of $C$ meet at the point $P$. The normal to the graph of $C$ at $x = 1$ passes through $P$ for certain values of $k$.

Show that these values of $k$ satisfy the equation $k^3-3k^2+k+2=0$.

(f) Hence, find the values of $k$ for which the normal to the graph of $C$ at $x = 1$ passes through $P$.

Consider the function f defined by $f(x) = \ln(x^2 - 3)$ for $x > \sqrt{3}$.

The following diagram shows part of the graph of f which crosses the x-axis at point A, with coordinates $(p, 0)$. The line L is the tangent to the graph of f at the point B.

(a) Find the exact value of $p$.

(b) Given that the gradient of L is $1$, find the x-coordinate of B.

The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = -2x + 5$ is the tangent to the graph of $f$ at $x = 1$.

Write down the value of $f'(1)$.

Find the value of $f(1)$.

The function $g$ is defined for all $x \in \mathbb{R}, x \neq 0$ where $g(x) = \frac{1}{x}$. The function $h$ is defined as $h(x) = f(g(x) )$.

Find the value of $h'(1)$.

Hence, find the equation of the tangent to the graph of $h$ at $x = 1$.

A function is defined by $f(x) = x^3 - 4x$.

(a) Find the equation of the tangent to the graph of $f$ at the point where $x = -1$.

(b) The tangent line found in part (a) intersects the graph of $f$ at a second point, P. Find the coordinates of P.

(c) Find the exact area of the finite region enclosed by the graph of $f$ and the tangent line.

A pharmaceutical company is testing a new drug. The concentration of the drug, $C$, in the bloodstream of a patient, in micrograms per millilitre ($\mu\text{g/mL}$), $t$ hours after administration, is modelled by the function $C(t) = 5te^{-0.2t}$, for $0 \le t \le 15$.

Sketch the graph of $C(t)$ for $0 \le t \le 15$, clearly indicating the coordinates of the initial concentration point $A$, the maximum concentration point $M$, and the concentration point $B$ at $t=15$ hours.

State the range of the concentration $C(t)$ during the observed period.

Find the equation of the straight line connecting the initial concentration point $A$ and the concentration point $B$ at $t=15$ hours.

Show that the rate of change of the drug concentration is given by $C'(t) = (5-t)e^{-0.2t}$.

At a certain time, the rate of change of the drug concentration is parallel to the line AB. Find the equation of the tangent line to the graph of $C(t)$ at this time. Give all coefficients in your equation correct to $3$ significant figures.

Calculate the area of the region enclosed by the graph of $C(t)$ and the line AB.

A function, $g$, has its derivative given by $g'(x) = -2x^2 + 8x + k$, where $k \in \mathbb{R}$. The following diagram shows part of the graph of $g'$.

The graph of $g'$ has an axis of symmetry $x = q$.

(a) Find the value of $q$.

(b) The vertex of the graph of $g'$ has a y-coordinate of 10. Find the value of $k$.

(c) Find the equation of the tangent to the graph of $g'$ at $x = 0$.

The graph of $g$ has a point of inflexion at $x = c$.

(d) (i) Find the value of $c$.

(ii) Find the values of $x$ for which the graph of $g$ is concave-up. Justify your answer.

The function $f$ is defined by $f(x) = e^x \sin(x)$, for $x \in [0, \pi]$.

(a) Show that the curve $y=f(x)$ has only one point of inflection in its domain, and determine its coordinates.

(b) Find the equations of the tangent and the normal to the curve at the point where $x = \pi$.

(c) Calculate the area of the triangle formed by this tangent, this normal, and the $y$-axis.

The trajectory of a small drone flying over a landscape can be modelled by the function $f(x) = -0.5x^2 + 4x + 1$, where $x$ is the horizontal distance in meters from the launch point and $f(x)$ is the altitude in meters. The drone passes through a checkpoint A at a horizontal distance of 2 meters.

(a) (i) Find the gradient of the tangent to the drone's trajectory at checkpoint A.

(a) (ii) Hence, write down the gradient of the normal to the drone's trajectory at checkpoint A.

(b) Write down the equation of the normal to the drone's trajectory at checkpoint A.

(c) A searchlight beam is directed along the normal line found in part (b). This beam intersects the drone's trajectory again at a second point B. Find the coordinates of B.