Chapter 2

What is \(a_{4}\) and what is the limit \(L\) ? After which \(N\) is \(\left|a_{n}-L\right|<\frac{1}{10} ?\) (Calculator allowed) (a) \(-1,+\frac{1}{2},-\frac{1}{3}, \ldots\) (b) \(\frac{1}{2}, \frac{1}{2}+\frac{1}{4}, \frac{1}{2}+\frac{1}{4}+\frac{1}{6}, \ldots\) (c) \(\frac{1}{2}, \frac{2}{4}, \frac{3}{8}, \ldots a_{n}=n / 2^{n}\) (d) \(1.1,1.11,1.111, \ldots\) (e) \(a_{n}=\sqrt[n]{n}\) (f) \(a_{n}=\sqrt{n^{2}+n}-n\) (g) \(1+1,\left(1+\frac{1}{2}\right)^{2},\left(1+\frac{1}{3}\right)^{3}, \ldots\)

Starting with \(f=x^{6},\) write down \(f^{\prime}\) and then \(f^{\prime \prime}\). (This is \(" f\) double prime," the derivative of \(f^{\prime} .\) ) After _______ derivatives of \(x^{6}\) you reach a constant. What constant?

a) Find the slope of \(y=12 / x\) (b) Find the equation of the tangent line at (2,6) . (c) Find the equation of the normal line at \((2,6) .\) (d) Find the equation of the secant line to (4,3) .

Find the derivatives of the functions in \(1-26\). $$ (x+1)(x-1) $$

Which of these ratios approach 1 as \(h \rightarrow 0 ?\) (a) \(\frac{h}{\sin h}\) (b) \(\frac{\sin ^{2} h}{h^{2}}\) (c) \(\frac{\sin h}{\sin 2 h}\) (d) \(\frac{\sin (-h)}{h}\)

Show by example that these statements are false: (a) If \(a_{n} \rightarrow L\) and \(b_{n} \rightarrow L\) then \(a_{n} / b_{n} \rightarrow 1\) (b) \(a_{n} \rightarrow L\) if and only if \(a_{n}^{2} \rightarrow L^{2}\) (c) If \(a_{n}<0\) and \(a_{n} \rightarrow L\) then \(L<0\) (d) If infinitely many \(a_{n}\) 's are inside every strip around zero then \(a_{n} \rightarrow 0\).

Find a function that has \(x^{6}\) as its derivative.

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{cl} \cos ^{3} x & x \neq \pi \\ c & x=\pi \end{array}\right. $$

For \(y=x^{2}+x\) find equations for (a) the tangent line and normal line at (1,2)\(;\) (b) the secant line to \(x=1+h, y=(1+h)^{2}+(1+h)\)

Find the derivatives of the functions in \(1-26\). $$ \left(x^{2}+1\right)\left(x^{2}-1\right) $$

Find \((\sin h) / h\) at \(h=0.5\) and 0.1 and .01 . Where does \((\sin h) / h\) go above \(.99 ?\)

Suppose \(f(x)=x^{2}\). Compute each ratio and set \(h=0\) (a) \(\frac{f(x+h)-f(x)}{h}\) (b) \(\frac{f(x+5 h)-f(x)}{5 h}\) (c) \(\frac{f(x+h)-f(x-h)}{2 h}\) (d) \(\frac{f(x+1)-f(x)}{h}\)

Which of these statements are equivalent to \(B \Rightarrow A\) ? (a) If \(A\) is true so is \(B\) (b) \(A\) is true if and only if \(B\) is true (c) \(B\) is a sufficient condition for \(A\) (d) \(A\) is a necessary condition for \(B\).

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ x^{2}+7 x+5 $$

A line goes through (1,-1) and (4,8) . Write its equation in point-slope form. Then write it as \(y=m x+b\).

Find the derivatives of the functions in \(1-26\). $$ \frac{1}{1+x}+\frac{1}{1+\sin x} $$

Find the limits as \(h \rightarrow 0\) of (a) \(\frac{\sin ^{2} h}{h}\) (b) \(\frac{\sin 5 h}{5 h}\) (c) \(\frac{\sin 5 h}{h}\) (d) \(\frac{\sin h}{5 h}\)

For \(f(x)=3 x\) and \(g(x)=1+3 x\), find \(f(4+h)\) and \(g(4+h)\) and \(f^{\prime}(4)\) and \(g^{\prime}(4)\). Sketch the graphs of \(f\) and \(g-\) why do they have the same slope?

Decide whether \(A \Rightarrow B\) or \(B \Rightarrow A\) or neither or both: (a) \(A=\left[a_{n} \rightarrow 1\right] \quad B=\left[-a_{n} \rightarrow-1\right]\) (b) \(A=\left[a_{n} \rightarrow 0\right] \quad B=\left[a_{n}-a_{n-1} \rightarrow 0\right]\) (c) \(A=\left[a_{n} \leqslant n\right] \quad B=\left[a_{n}=n\right]\) (d) \(A=\left[a_{n} \rightarrow 0\right] \quad B=\left[\sin a_{n} \rightarrow 0\right]\) (e) \(A=\left[a_{n} \rightarrow 0\right] \quad B=\left[1 / a_{n}\right.\) fails to converge \(]\) (f) \(A=\left[a_{n}

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ 1+(7 / x)+\left(5 / x^{2}\right) $$

The tangent line to \(y=x^{3}+6 x\) at the origin is \(y=\) Does it cross the curve again?

If the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) approaches zero, prove that we can put those numbers in any order and the new sequence still approaches zero.

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ 1+x+x^{2}+x^{3}+x^{4} $$

For \(f(x)=1 / x,\) sketch the graphs of \(f(x)+1\) and \(f(x+1)\). Which one has the derivative \(-1 / x^{2} ?\)

Suppose \(f(x) \rightarrow L\) and \(g(x) \rightarrow M\) as \(x \rightarrow a\). Prove from the definitions that \(f(x)+g(x) \rightarrow L+M\) as \(x \rightarrow a\)

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ \left(x^{2}+1\right)^{2} $$

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{rl} x^{3} & x \neq c \\ -8 & x=c \end{array}\right. $$

Find the derivatives of the functions in \(1-26\). $$ (x-1)^{2}(x-2)^{2} $$

Choose \(c\) so that the line \(y=x\) is tangent to the parabola \(y=x^{2}+c .\) They have the same slope where they touch.

Find the limits if they exist. An \(\varepsilon-\delta\) test is not required. $$ \lim _{t \rightarrow 2} \frac{t+3}{t^{2}-2} $$

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ x^{n}+x^{-n} $$

Find the derivatives of the functions in \(1-26\). $$ x^{2} \cos x+2 x \sin x $$

For \(y=x^{2}\) the secant line from \(\left(a, a^{2}\right)\) to \(\left(c, c^{2}\right)\) has the equation \(\longrightarrow\) Do the division by \(c-a\) to find the tangent line as \(c\) approaches \(a\).

The key to trigonometry is \(\cos ^{2} \theta=1-\sin ^{2} \theta\). Set \(\sin \theta \approx \theta\) to find \(\cos ^{2} \theta \approx 1-\theta^{2}\). The square root is \(\cos \theta \approx 1-\frac{1}{2} \theta^{2}\). Reason: Squaring gives \(\cos ^{2} \theta \approx\) and the correction term is very small near \(\theta=0\).

Find the limits if they exist. An \(\varepsilon-\delta\) test is not required. $$ \lim _{t \rightarrow 2} \frac{t^{2}+3}{t-2} $$

Construct a function that has the same slope at \(x=1\) and \(x=2\). Then find two points where \(y=x^{4}-2 x^{2}\) has the same tangent line (draw the graph).

(Calculator) Compare \(\cos \theta\) with \(1-\frac{1}{2} \theta^{2}\) for (a) \(\theta=0.1\) (b) \(\theta=0.5\) (c) \(\theta=30^{\circ}\) (d) \(\theta=3^{\circ}\).

If \(f(t)=1 / t,\) what is the average velocity between \(t=\frac{1}{2}\) and \(t=2 ?\) What is the average between \(t=\frac{1}{2}\) and \(t=1 ?\) What is the average (to one decimal place) between \(t=\frac{1}{2}\) and \(t=101 / 200 ?\)

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ 1+x+\frac{1}{2} x^{2}+\frac{1}{6} x^{3}+\frac{1}{24} x^{4} $$

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{cc} (\sin x) / x^{2} & x \neq 0 \\ c & x=0 \end{array}\right. $$

Find \(\Delta y / \Delta x\) for \(y(x)=x+x^{2}\). Then find \(d y / d x\).

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ \frac{2}{3} x^{3 / 2}+\frac{2}{5} x^{5 / 2} $$

Find the derivatives of the functions in \(1-26\). $$ \frac{x^{2}+1}{x^{2}-1}+\frac{\sin x}{\cos x} $$

Find the limits as \(h \rightarrow 0\) : (a) \(\frac{1-\cos h}{h^{2}}\) (b) \(\frac{1-\cos ^{2} h}{h^{2}}\) (c) \(\frac{1-\cos ^{2} h}{\sin ^{2} h}\) (d) \(\frac{1-\cos 2 h}{h}\)

Find \(\Delta y / \Delta x\) and \(d y / d x\) for \(y(x)=1+2 x+3 x^{2}\).

Name two functions with \(d f / d x=1 / x^{2}\).

Find the derivatives of the functions in \(1-26\). $$ x^{1 / 2} \sin ^{2} x+(\sin x)^{1 / 2} $$

Find by calculator or calculus: (a) \(\lim _{h \rightarrow 0} \frac{\sin 3 h}{\sin 2 h}\) (b) \(\lim _{h \rightarrow 0} \frac{1-\cos 2 h}{1-\cos h}\).

What are the equations of the tangent line and normal line to \(y=\sin x\) at \(x=\pi / 2 ?\)

When \(f(t)=4 / t,\) simplify the difference \(f(t+\Delta t)-f(t)\) divide by \(\Delta t,\) and set \(\Delta t=0 .\) The result is \(f^{\prime}(t)\).