Exercises: Tangent line to a curve

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 $y - f(1) = f'(1) \times (x - 1)$.

Determine the equation of the tangent line to the function $f(x) = x^3 - 2x$ at the point with x-coordinate $x = 1$.

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 $y = 4x - 1$.

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 $x = 1$.

A cyclist training in a velodrome travels $s(t) = 2t^2 + 3t$ meters during the first $t$ seconds.

1. a) Find the equation of the tangent line to the graph of $s(t)$ at time $t = 2$.

1. b) What physical quantity does the slope of this tangent line represent? State its value and its units.

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 very close to $1$?

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.

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

1. a) Find the equation of the tangent line at the point with x-coordinate $x = 3$.

1. b) Determine the points of intersection of this tangent line with the coordinate axes.

1. c) Calculate the area of the triangle formed by the tangent line and the two axes.

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 $x = 1$.

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.

1. a) Write down the equation of the tangent line to the parabola at a general point with x-coordinate $x = a$.

1. 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 tangent line at a general point with x-coordinate $x = a$ and impose the condition that it passes through $(0, 0)$.)

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.

1. a) Verify that the two curves intersect at the points with x-coordinates $x = -1$ and $x = 1$.

1. b) Calculate the gradients of the tangents to both curves at the point with x-coordinate $x = 1$.

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.

1. a) Find the equation of the tangent line to $f(x) = \sqrt{x}$ at the point with x-coordinate $x = 100$.

1. b) Use this line to give an estimate of $\sqrt{102}$ to two decimal places.

1. 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.