DP Mathematics: Analysis and Approaches Questionbank

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

Find the least value of $n$.

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

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

Find $\text{OA}$.

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

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

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

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

Find $\text{BC}$.

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

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 $x$.

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

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 $m$.

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.

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

Find the value of $k$.

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

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

Find the x-coordinate of P.

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

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

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 $C$.

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

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 $f\left(x\right)$.

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

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

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 $\int x{\text{e}}^{x^2-1}\text{d}x$.

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

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

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

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.

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.

Let $f^{\prime }(x)={\sin }^3(2x)\cos (2x)$. Find $f(x)$, given that $f\left(\frac{\pi }{4}\right)=1$.

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 $f(x)$.

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

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

Hence find the exact area of $R$.

Find the initial displacement of particle A from point P.

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

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

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

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

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

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

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

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

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.