Find the value of $t$ when particle A first changes direction.

Find the value of $k$ and the value of $p$.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt=\ln \frac{P}{P_0}\left(\frac{N-P_0}{N-P}\right)$.

Let $f\left(x\right)=\frac{6-2x}{\sqrt{16+6x-x^2}}$. The following diagram shows part of the graph of $f$.

The region R is enclosed by the graph of $f$, the $x$-axis, and the $y$-axis. Find the area of R.

Write down the first value of $t$ at which P changes direction.

Find the total distance travelled by P, for $0⩽t⩽8$.

Let $f^{\prime }(x)=\frac{3x^2}{{(x^3+1)}^5}$. Given that $f(0)=1$, find $f(x)$.

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

A function $f$ satisfies the conditions $f\left(0\right)=-4$, $f\left(1\right)=0$ and its second derivative is $f^{″}\left(x\right)=15\sqrt{x}+\frac{1}{{\left(x+1\right)}^2}$, $x$ ≥ 0.

Find $f\left(x\right)$.

Find the total distance travelled by particle A in the first 3 seconds.

Write down an expression for the area of $R$.

Find $\int {\left(f\left(x\right)\right)}^2\text{d}x$.

Find $\int x{\text{e}}^{x^2-1}\text{d}x$.

Hence, find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

Show that ${\int }_0^{m}f\left(x\right)dx=\frac{32}{7}{\sin }^7 m$, where $m$ is a positive real constant.

It is given that ${\int }_m^{\frac{\mathrm{\pi }}{2}}f\left(x\right)dx=\frac{127}{28}$, where $0\le m\le \frac{\mathrm{\pi }}{2}$. Find the value of $m$.

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.

Given that particles A and B start at the same point, find the displacement function $s_{\text{B}}$ for particle B.

Find $\text{E}\left(X\right)$.

The graph of a function $f$ passes through the point $\left(\ln 4, 20\right)$.

Given that $f\text{'}\left(x\right)=6{\text{e}}^{2x}$, find $f\left(x\right)$.

Show that $f\text{'}\text{'}\left(x\right)=\frac{24-6x^2}{{\left(x^2+4\right)}^2}$.

Hence find the exact area of $R$.

Find the initial displacement of particle A from point P.

Find the least value of $n$.

Find the other value of $t$ when particles A and B meet.

State the cross-sectional radius of the bowl at this point.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

The region R is enclosed by the graph of $f$, the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of $k$.

Show that the area of the shaded region is $12\mathrm{\pi }$.

Find the x-coordinate of P.

Show that the x-coordinate of Q is $\frac{\text{1}}{5}{\text{e}}^{\frac{3}{2}}$.

Find $\int \frac{6x}{x^2+4}\text{d}x$.

Find the distance that the rocket travels during the first stage.

Find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

Find the value of $k$.

Find the value of $t$ when particle A first reaches point P.

Given that $2x^3-3x+1$ can be expressed in the form $Ax\left(x^2+1\right)+Bx+C$, find the values of the constants $A$, $B$ and $C$.

Find $\text{OA}$.

Given that ${\int }_0^{\text{ln}k}{\text{e}}^{2x}\text{d}x=12$, find the value of $k$.

Hence find $\int \frac{2x^3-3x+1}{x^2+1}\text{d}x$.

Find an expression for the velocity, $v$ m s⁻¹, of the rocket during the first stage.

The derivative of a function $f$ is given by $f^{\prime }(x)=2{\text{e}}^{-3x}$. The graph of $f$ passes through $\left(\frac{1}{3}\text{,}5\right)$.

Find $f(x)$.

Let $f^{\prime }\left(x\right)=\frac{8x}{\sqrt{2x^2+1}}$. Given that $f\left(0\right)=5$, find $f\left(x\right)$.

Hence find $\int \left(3x^2+1\right)\sqrt{x^3+x}\text{d}x$.

Show that $f^{\prime }\left(x\right)=\frac{1-\text{ln}5x}{k{x}^2}$.

A second particle Q also moves along a straight line. Its velocity, $v_{\text{Q}}\text{ m}{\text{s}}^{-1}$ after $t$ seconds is given by $v_{\text{Q}}=\sqrt{t}$ for $0⩽t⩽8$. After $k$ seconds Q has travelled the same total distance as P.

Find $k$.

Find $f(x)$, given that $f^{\prime }(x)=x{\text{e}}^{x^2-1}$ and $f(-1)=3$.

Let $R$ be the region enclosed by the graph of $f$, the $x$-axis and the lines $x=1$ and $x=3$. The area of $R$ is $19.6$, correct to three significant figures.

Find $f\left(x\right)$.

The derivative of a function $f$ is given by $f\text{'}\left(x\right)=3\sqrt{x}$.

Given that $f\left(1\right)=3$, find the value of $f\left(4\right)$.

Show that $\sqrt{16+k}-\sqrt{k}=\sqrt{k}\sqrt{16+k}$.

Show that the volume of the solid formed is $\frac{15k^2\mathrm{\pi }}{34}$ cubic units.

Find $\text{BC}$.

Find the value of $k$ that satisfies the requirements of Pedro’s design.

Given that $\frac{dy}{dx}=\cos \left(x-\frac{\mathrm{\pi }}{4}\right)$ and $y=2$ when $x=\frac{3\mathrm{\pi }}{4}$, find $y$ in terms of $x$.