Topic 5.4: Tangents and Normals Questions

The graphs of $y=e^x$ and $y=e^{-x}$ cross at one point. Show that they cross at right angles there, by working out both slopes at that point.

Write down the gradient of the normal at that point.

Let $y=\frac{1}{x^2}$. Show that the normal to this curve at the point where $x=t$ has gradient $\frac{t^3}{2}$.

The curve $y=\frac{4}{x}$ passes through the point $(2,2)$. Show that the normal to the curve at this point is the line $y=x$.

For the curve y = x² − 2x, find where the tangent has gradient 0 and state the equation of that tangent.

The tangent to a curve at a point has gradient 5. Find the gradient of the normal there.

Write down the gradient of the tangent at $x=2$.

The tangent to the curve $y=g(x)$ at the point where $x=-1$ is the line $y=3x+7$. Find $g(-1)$ and $g^{\prime }(-1)$.

Show that the normal at that point has gradient $-\frac{1}{2}$.

The tangent to the curve $y=f(x)$ at the point where $x=3$ has equation $y=5-2x$. Write down the value of $f^{\prime }(3)$.

The function f is defined by f(x) = 2x³ − 4x² + 2x − 5. The tangent to the graph of f at x = 1 is horizontal. The graph of f intersects the y-axis at the point P.

Show that the tangent to the graph of f at x = 1 passes through P.

A curve has equation $y=x^3-4x$. Calculate the gradient of its tangent when $x=2$.

For the curve $y=x^2-2x$, find where the tangent has gradient 0 and state the equation of that tangent.

The function f is defined by f(x) = x³ + 4x² + 4x + 3. The tangent to the graph of f at x = −2 is horizontal. The graph of f intersects the y-axis at the point P.

Show that the tangent to the graph of f at x = −2 passes through P.

The curve $y=x^3-3x$ has $\frac{dy}{dx}=3x^2-3$. Write down the coordinates of one point on the curve where the tangent is horizontal.

Find the tangent to $y=\frac{6}{x}$ where $x=3$, giving your answer in the form $y=mx+c$.

The curve $y=x^2-6x$ has a horizontal tangent. Write down its equation.

The tangent to $y=x^2-7x$ is parallel to the line $y=5x+1$. Find the point of contact.

Show that the gradient of the tangent at $x=1$ is $7$.

Find the equation of the tangent to the curve at the point where $x=2$.