Chapter 5

The antiderivative of \(d^{2} f / d x^{2}\) is \(d f / d x\). What is the sum \(\left(f_{2}-2 f_{1}+f_{0}\right)+\left(f_{3}-2 f_{2}+f_{1}\right)+\cdots+\left(\int_{9}-2 f_{8}+f_{7}\right) ?\)

Choose \(u(x)\) in \(11-18\) and change limits. Compute the integral in \(11-16\) $$ \int_{0}^{1} x^{3}(1-x)^{3} d x(u=1-x) $$

Find the indefinite integrels in \(1-20\). \(\int \sec ^{2} x \tan ^{2} x d x\)

About a company whose expense rate \(t(x)=6-x\) is decreasing. The rectangles of heights 6, 5, 4, 3, 2, 1 give a total estimated expense of _________. Draw them enclosing the triangle to show why this total is too high.

Write down the three equations \(A y(0)+B y\left(\frac{1}{2}\right)+C y(1)=1\) for the three integrals \(I=\int_{0}^{1} 1 d x, \int_{0}^{1} x d x, \int_{0}^{1} x^{2} d x\). Solve for \(A, B, C\) and name the rule.

Find the derivatives of the following functions \(F(x)\). $$ \int_{0}^{\sin x} \sin ^{-1} t d t $$

Draw \(y=\sin x\) from 0 to \(\pi .\) Three rectangles (base \(\pi / 3)\) and six rectangles (base \(\pi / 6)\) contain an arch of the sine function. Find the areas and guess the limit.

Find the indefinite integrels in \(1-20\). \(\int \cos x \tan x d x\)

Prove by induction: \(1+3+\cdots+(2 n-1)=n^{2}\)

(a) Which property of integrals proves \(\int_{0}^{1} v(x) d x \leqslant\) \(\int_{0}^{1}|v(x)| d x ?\) (b) Which property proves \(-\int_{0}^{1} v(x) d x \leqslant \int_{0}^{1}|v(x)| d x\) ? Together these are Property \(\mathbf{8}:\left|\int_{0}^{1} b(x) d x\right| \leqslant \int_{0}^{1}|v(x)| d x\).

Find the indefinite integrels in \(1-20\). \(\int \sin ^{3} x d x\)

Verify that \(1^{3}+2^{3}+\cdots+n^{3}\) is \(f_{n}=\frac{1}{4} n^{2}(n+1)^{2}\) by checking \(f_{1}\) and \(f_{x}-f_{n-1}\). The text has a proof without words.

True or false (a) If \(d f / d x=d g / d x\) then \(f(x)=g(x)\). (b) If \(d^{2} f / d x^{2}=d^{2} g / d x^{2}\) then \(f(x)=g(x)+C\). (c) If \(3>x\) then the derivative of \(\int_{3}^{x} v(t) d t\) is \(-v(x)\). (d) The derivative of \(\int_{1}^{3} v(x) d x\) is zero.

Draw \(y=1 / x^{2}\) for \(0

In \(21-32\) find a function \(y(x)\) that solves the differential equation. \(d y / d x=x^{2}+\sqrt{x}\)

Suppose \(f_{n}\) has the form \(a n+b n^{2}+c n^{3}\). If you know \(f_{1}=1, f_{2}=5, f_{3}=14,\) turn those into three equations for \(a, b, c .\) The solutions $a=\frac{1}{6}, b=\frac{1}{2}, c=\$$ give what formula?

For \(F(x)=\int_{x}^{2 x} \sin t d t,\) locate \(F(\pi+\Delta x)-F(\pi)\) on a sine graph. Where is \(F(\Delta x)-F(0)\) ?

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=y^{2} \quad\left(\right.\) try \(\left.y=c x^{n}\right)\)

About a company whose expense rate \(t(x)=6-x\) is decreasing. What is the area \(f(x)\) under the line \(t(x)=6-x\) above the interval from \(x\) to 6 ? What is the derivative of this \(f(x)\) ?

The graph of \(y(x)=1 /\left(x^{2}+10^{-10}\right)\) has a sharp spike and a long tail. Estimate \(\int_{0}^{1} y d x\) from \(T_{10}\) and \(T_{100}\) (don't expect much). Then substitute \(x=10^{-5} \tan \theta, d x=10^{-5} \sec ^{2} \theta d \theta\) and integrate \(10^{3}\) from 0 to \(\pi / 4\).

Find the function \(v(x)\) whose average value between 0 and \(x\) is \(\cos x .\) Start from \(\int_{0}^{x} v(t) d t=x \cos x\).

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=\sqrt{1-2 x}\)

Compute \(\int_{0}^{4}|x-\pi| d x\) from \(T_{4}\) and compare with the divide and conquer method of separating \(\int_{0}^{1}|x-\pi| d x\) from \(\int_{x}^{4}|x-\pi| d x\).

If the positive numbers \(v_{n}\) approach zero as \(n \rightarrow \infty\) prove that their average \(\left(v_{1}+\cdots+v_{n}\right) / n\) also approaches zero.

Suppose \(d f / d x=2 x\). We know that \(d\left(x^{2}\right) / d x=2 x\). How do we prove that \(f(x)=x^{2}+C ?\)

Find the area under the parabola \(v=x^{2}\) from \(x=0\) to \(x=4 .\) Relate it to the area \(16 / 3\) below \(\sqrt{x}\).

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=1 / \sqrt{1-2 x}\)

Add \(n=50\) to the table for \(S_{n}=1^{2}+\cdots+n^{2}\) and compute \(E_{\mathrm{s} 0}\). Find an approximate formula for \(E_{n}\).

Find \(a, b, c\) so that \(y=a x^{2}+b x+c\) equals 1,3,7 at \(x=0, \frac{1}{2}, 1\) (three equations). Check that \(\frac{1}{6} \cdot 1+\frac{4}{6} \cdot 3+\frac{1}{6} \cdot 7\) equals \(\int_{0}^{1} y d x\).

Find the average distance from \(x=a\) to points in the interval \(0 \leqslant x \leqslant 2\). Is the formula different if \(a<2 ?\)

If \(\int_{-x}^{0} v(t) d t=\int_{0}^{x} v(t) d t\) (equal areas left and right of zero), then \(v(x)\) is an ________ function. Take derivatives to prove it.

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=1 / y\)

Find \(c\) in \(S-I=c(\Delta x)^{4}\left[y^{\prime \prime}(1)-y^{\prime \prime \prime}(0)\right]\) by taking \(y=x^{4}\) and \(\Delta x=1\)

Example 2 said that \(\int_{2 x}^{3 x} d t / t\) does not really depend on \(x\) (or \(t !)\). Substitute \(x u\) for \(t\) and watch the limits on \(u\).

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=x / y\)

Compute the area of 208 rectangles under \(u(x)=\sqrt{x}\) from \(x=0\) to \(x=4\)

Find \(c\) in \(G-I \Leftrightarrow c(\Delta x)^{4}\left[y^{\prime \prime \prime}(1)-y^{\prime \prime \prime}(-1)\right]\) by taking \(y=x^{4}, \Delta x=2,\) and \(G=(-1 / \sqrt{3})^{4}+(1 / \sqrt{3})^{4}\)

A point \(P\) is chosen randomly along a semicircle (see figure: equal probability for equal arcs). What is the average distance \(y\) from the \(x\) axis? The radius is \(1 .\)

True or false, with reason: (a) All continuous functions have derivatives. (b) All continuous functions have antiderivatives. (c) All antiderivatives have derivatives. (d) \(A(x)=\int_{2 x}^{3 x} d t / t^{2}\) has \(d A / d x=0\).

Find a function \(y(x)\) that solves the differential equation. \(d^{2} y / d x^{2}=1\)

Show that \(|1-5|<|1|+|-5|\). Always \(\left|v_{1}+v_{2}\right|<\left|v_{1}\right|+\left|v_{2}\right|\) unless

What condition on \(y(x)\) makes \(L_{n}=R_{n}=T_{n}\) for the integral \(\int_{a}^{b} y(x) d x ?\)

Find a function \(y(x)\) that solves the differential equation. \(d^{5} y / d x^{5}=1\)

(A classic way to compute \(\pi)\) A \(2^{\prime \prime}\) needle is tossed onto a floor with boards \(2^{\prime \prime}\) wide. Find the probability of falling across a crack. (This happens when \(\cos \theta>y=\) distance from midpoint of needle to nearest crack. In the rectangle \(0 \leqslant \theta \in \pi / 2,0 \leqslant y \leqslant 1,\) shade the parl where \(\cos \theta>y\) and find the fraction of area that is shaded.)

Find \(\int_{1}^{x} v(t) d t\) from the facts. $$ \int_{0}^{x} v(t) d t=\frac{x}{x+2}. $$

The dowble swm \(\sum_{i=1}^{2}\left[\sum_{j=1}^{3}(i+j)\right]\) is \(v_{1}=\sum_{i=1}^{3}(1+j)\) plus \(v_{2}=\sum_{j=1}^{3}(2+j)\). Compute \(v_{1}\) and \(v_{2}\) and the double sum.

If \(d f / d x=-v(x)\) and \(f(1)=0,\) explain \(f(x)=\int_{x}^{1} v(t) d t\).

The double sum \(\sum_{i=1}^{2}\left(\sum_{j=1}^{3} w_{i, j}\right)\) is \(\left(w_{1,1}+w_{1,2}+w_{1,3}\right)+\) The double sum \(\sum_{j=1}^{3}\left(\sum_{i=1}^{2} w_{i, j}\right)\) is \(\left(w_{1,1}+w_{2,1}\right)+\left(w_{1,2}+w_{2,2}\right)+\) Compare.

The cube \(0 \leqslant x, y, z \leqslant s\) has volume \(V=\) ______. The three square faces with \(x=s\) or \(y=s\) or \(z=s\) have total area \(A=\) _______. If \(s\) is increased by \(\Delta s\), the extra volume has the shape of ___________. That volume \(\Delta V\) is approximately _______ .So \(d V / d s=\) _______.

If you choose a random point in the square \(0 \leqslant x \leqslant 1\), \(0 \leqslant y \leqslant 1,\) what is the chance that its coordinates have \(y^{2} \leqslant x ?\)