Chapter 4

Give a reasonable interpretation of the formula $$ \frac{d}{d x}(f+g)=\frac{d f}{d x}+\frac{d g}{d x} $$

Suppose that the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are differentiable and define \(h \equiv f \circ g: \mathbb{R} \rightarrow \mathbb{R} .\) Find \(h^{\prime}(1)\) and \(h^{\prime}(2)\) if $$ g(1)=2, g(2)=1, f^{\prime}(1)=-1, f^{\prime}(2)=2, g^{\prime}(1)=3, g^{\prime}(2)=4 $$

For each of the following statements, determine whether it is true or false and justify your answer. a. If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous at \(x_{0},\) then it is differentiable at \(x_{0}\). b. If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x_{0},\) then it is continuous at \(x_{0}\). c. The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable if the function \(f^{2}: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable.

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has two derivatives, with \(f(0)=f^{\prime}(0)=0\) and \(\left|f^{\prime \prime}(x)\right| \leq 1\) if \(|x| \leq 1\). Prove that \(f(x) \leq 1 / 2\) if \(|x| \leq 1\).

Give a reasonable interpretation of the formula $$ \frac{d f}{d r}=\frac{d f}{d u} \cdot \frac{d u}{d s} \cdot \frac{d s}{d r} $$

Define \(f(x)=x^{3}+2 x+1\) for all \(x .\) Find the equation of the tangent line to the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}\) at the point (2,13).

Let \(p: \mathbb{R} \rightarrow \mathbb{R}\) be a polynomial of degree no greater than \(5 .\) Suppose that at some point \(x_{0}\) in \(\mathbb{R}\) $$ p\left(x_{0}\right)=p^{\prime}\left(x_{0}\right)=\cdots=p^{(5)}\left(x_{0}\right)=0 $$ Prove that \(p(x)=0\) for all \(x\) in \(\mathbb{R}\).

Define \(f(t)=t^{2}\) for \(0 \leq t \leq 1\) and \(g(t)=t^{3}\) for \(0 \leq t \leq 1\) a. Find the number \(c\) with \(0 < c < 1\) at which $$ \frac{f(1)-f(0)}{g(1)-g(0)}=\frac{f^{\prime}(c)}{g^{\prime}(c)} $$ b. Show that there does not exist a number \(c\) with \(0

Define \(f(x)=1 /(1+x)\) for \(x\) in \(I \equiv(0,1)\). Show that \(f: I \rightarrow \mathbb{R}\) is strictly decreasing and differentiable and that \(f(I)=(1 / 2,1) \equiv J .\) Show that \(f^{-1}(y)=\) \((1-y) / y\) for \(y\) in \(J .\) Calculate the derivative of the inverse directly and then check that this calculation agrees with formula (4.6).

For \(c>0\), prove that the following equation does not have two solutions: $$ x^{3}-3 x+c=0, \quad 0

Suppose that the functions \(f:[a, b] \rightarrow \mathbb{R}\) and \(g:[a, b] \rightarrow \mathbb{R}\) are continuous and that their restrictions to the open interval \((a, b)\) are differentiable. Also suppose that \(\left|f^{\prime}(x)\right| \geq\left|g^{\prime}(x)\right| > 0\) for all \(x\) in \((a, b) .\) Prove that \(|f(u)-f(v)| \geq|g(u)-g(v)| \quad\) for all \(u, v\) in \([a, b]\)

Use the definition of derivative to compute the derivative of the following functions at \(x=1:\) a. \(f(x)=\sqrt{x+1}\) for all \(x>0\) b. \(f(x)=x^{3}+2 x\) for all \(x\). c. \(f(x)=1 /\left(1+x^{2}\right)\) for all \(x\).

Let \(I\) be a neighborhood of \(x_{0}\) and let \(f: I \rightarrow \mathbb{R}\) be continuous, strictly monotone, and differentiable at \(x_{0}\). Assume that \(f^{\prime}\left(x_{0}\right)=0 .\) Use the characteristic property of inverses, $$ f^{-1}(f(x))=x $$ for \(x\) in \(I\), and the Chain Rule to prove that the inverse function \(f^{-1}: f(I) \rightarrow \mathbb{R}\) is not differentiable at \(f\left(x_{0}\right) .\) Thus, the assumption in Theorem 4.11 that \(f^{\prime}\left(x_{0}\right) \neq 0\) is necessary.

Prove that the following equation has exactly one solution: $$ x^{5}+5 x+1=0, \quad-1

Suppose that the function \(f:(-1,1) \rightarrow \mathbb{R}\) has \(n\) derivatives and that its \(n\) th derivative \(f^{(n)}:(-1,1) \rightarrow \mathbb{R}\) is bounded. Assume also that $$ f(0)=f^{\prime}(0)=\cdots=f^{(n-1)}(0)=0 $$ Prove that there is a positive number \(M\) such that \(|f(x)| \leq M|x|^{n} \quad\) for all \(x\) in (-1,1)

Evaluate the following limits or determine that they do not exist: a. \(\lim _{x \rightarrow 0} \frac{x^{2}}{x}\) b. \(\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}\) c. \(\lim _{x \rightarrow 0} \frac{x-1}{\sqrt{x}-1}\) d. \(\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}\)

Suppose that the function \(f:(0, \infty) \rightarrow \mathbb{R}\) is differentiable and let \(c>0 .\) Now define \(g:(0, \infty) \rightarrow \mathbb{R}\) by \(g(x)=f(c x)\) for \(x>0 .\) Just using the definition of derivative, show that \(g^{\prime}(x)=c f^{\prime}(c x)\) for \(x>0\).

Suppose that the function \(f:(-1,1) \rightarrow \mathbb{R}\) has \(n\) derivatives. Assume that there is a positive number \(M\) such that $$ |f(x)| \leq M|x|^{n} $$ for all \(x\) in (-1,1) $$ \text { Prove that } f(0)=f^{\prime}(0)=\cdots=f^{(n-1)}(0)=0 $$

Suppose that the function \(h: \mathbb{R} \rightarrow \mathbb{R}\) is strictly monotone differentiable, \(h^{\prime}(x)>0\) for all \(x,\) and \(h(\mathbb{R})=\mathbb{R} .\) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be differentiable and define \(g(x)=\) \(f\left(h^{-1}(x)\right)\) for all \(x\). Find \(g^{\prime}(x)\)

For any numbers \(a\) and \(b\) and an even natural number \(n,\) show that the following equation has at most two solutions: $$ x^{n}+a x+b=0, \quad x \text { in } \mathbb{R} $$ Is this true if \(n\) is odd?

Let \(I\) be a neighborhood of \(x_{0}\) and suppose that the function \(f: I \rightarrow \mathbb{R}\) has two continuous derivatives. Prove that $$ \lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-2 f\left(x_{0}\right)+f\left(x_{0}-h\right)}{h^{2}}=f^{\prime \prime}\left(x_{0}\right) $$

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that \(\left\\{x_{n}\right\\}\) is a strictly increasing bounded sequence with \(f\left(x_{n}\right) \leq f\left(x_{n+1}\right)\) for all \(n\) in \(\mathbb{N} .\) Prove that there is a number \(x_{0}\) at which \(f^{\prime}\left(x_{0}\right) \geq 0 .\) (Hint: Apply the Monotone Convergence Theorem.)

Let \(I\) be an open interval and \(n\) be a natural number. Suppose that both \(f: I \rightarrow \mathbb{R}\) and \(g: I \rightarrow \mathbb{R}\) have \(n\) derivatives. Prove that \(f g: I \rightarrow \mathbb{R}\) has \(n\) derivatives, obtaining the following formula called Leibnitz's formula: \((f g)^{(n)}(x)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}\right) f^{(k)}(x) g^{(n-k)}(x) \quad\) for all \(x\) in \(I .\) Write the formula out explicitly for \(n=2\) and \(n=3\)

A function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is called even if $$ f(x)=f(-x) \quad \text { for all } x $$ and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is called odd if $$ f(x)=-f(-x) \quad \text { for all } x $$ Prove that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and odd, \(f^{\prime}: \mathbb{R} \rightarrow \mathbb{R}\) is even.

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has the property that $$-x^{2} \leq f(x) \leq x^{2}$$ for all \(x\). Prove that \(f\) is differentiable at \(x=0\) and that \(f^{\prime}(0)=0\).

Show that there does not exist a differentiable function \(F: \mathbb{R} \rightarrow \mathbb{R},\) with \(F^{\prime}(x)=0\) if \(x<0\) and \(F^{\prime}(x)=1\) if \(x \geq 0\), by arguing that such a function would necessarily be (i) continuous, (ii) constant on \((-\infty, 0),\) and (iii) of the form \(F(x)=A+B x\) on \((0, \infty),\) and then deriving a contradiction.

Let \(n\) be a natural number. Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that the following equation has at most \(n-1\) solutions: $$ f^{\prime}(x)=0, \quad x \text { in } \mathbb{R} $$ Prove that the following equation has at most \(n\) solutions: $$ f(x)=0, \quad x \text { in } \mathbb{R} $$

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x=0 .\) Also, suppose that for each natural number \(n, g(1 / n)=0 .\) Prove that \(g(0)=0\) and \(g^{\prime}(0)=0 .\)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and monotonically increasing. Show that \(f^{\prime}(x) \geq 0\) for all \(x\) .

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that $$ \left\\{\begin{array}{ll} f^{\prime}(x)=x+x^{3}+2 & \text { for all } x \text { in } \mathbb{R} \\ f(0)=5 \end{array}\right. $$ What is the function \(f: \mathbb{R} \rightarrow \mathbb{R} ?\)

Suppose that the function \(g:(-1,1) \rightarrow \mathbb{R}\) is differentiable and that $$ \left\\{\begin{array}{l} g^{\prime}(x)=x / \sqrt{1-x^{2}} \quad \text { for }-1

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x_{0} .\) Analyze the limit $$\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}-h\right)}{h}$$

Let \(g: \mathbb{R} \rightarrow \mathbb{R}\) and \(f: \mathbb{R} \rightarrow \mathbb{R}\) be differentiable functions and suppose that $$ g(x) f^{\prime}(x)=f(x) g^{\prime}(x) \quad \text { for all } x $$ If \(g(x) \neq 0\) for all \(x\) in \(\mathbb{R},\) show that there is some \(c\) in \(\mathbb{R}\) such that \(f(x)=\operatorname{cg}(x)\) for all \(x\) in \(\mathbb{R}\).

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are each differentiable and that $$ \left\\{\begin{array}{l} f^{\prime}(x)=g(x) \text { and } \quad g^{\prime}(x)=-f(x) \quad \text { for all } x \\ f(0)=0 \quad \text { and } \quad g(0)=1 \end{array}\right. $$ Prove that $$ [f(x)]^{2}+[g(x)]^{2}=1 $$ for all \(x\). (Hint: Define \(h(x) \equiv[f(x)]^{2}+[g(x)]^{2}\) for all \(x\). Show that \(h: \mathbb{R} \rightarrow \mathbb{R}\) is a constant function.)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(0 .\) For real numbers \(a, b,\) and \(c,\) with \(c \neq 0,\) prove that $$\lim _{x \rightarrow 0} \frac{f(a x)-f(b x)}{c x}=\left[\frac{a-b}{c}\right] f^{\prime}(0)$$.

Let the function \(h: \mathbb{R} \rightarrow \mathbb{R}\) be bounded. Define the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$f(x)=1+4 x+x^{2} h(x)$$ for all \(x\). Prove that \(f(0)=1\) and \(f^{\prime}(0)=4\). (Note: There is no assumption about the differentiability of the function \(h.)\)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable, that \(f^{\prime}: \mathbb{R} \rightarrow \mathbb{R}\) is continuous at \(0,\) and that \(f^{\prime}(0)>0 .\) Prove that there is an open interval \(I\) containing 0 such that \(f: I \rightarrow \mathbb{R}\) is strictly monotonic.

For a natural number \(n,\) the Geometric Sum Formula asserts that $$1+x+\cdots+x^{n}=\frac{1-x^{n+1}}{1-x} \quad \text { if } x \neq 1$$ By differentiating, find a formula for $$1+x+2 x^{2}+\cdots+n x^{n}$$ and then for $$1^{2}+2^{2} x+\cdots+n^{2} x^{n-1}$$.

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) have the property that there is a positive number \(c\) such that \(|f(u)-f(v)| \leq c(u-v)^{2}\) for all \(u, v\) in \(\mathbb{R} .\) Prove that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is constant.

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that there is a positive number \(c\) such that $$ f^{\prime}(x) \geq c $$ for all \(x\). Prove that $$ f(x) \geq f(0)+c x \text { if } x \geq 0 \quad \text { and } \quad f(x) \leq f(0)+c x \text { if } x \leq 0 $$ Use these inequalities to prove that \(f(\mathbb{R})=\mathbb{R}\).

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) have two derivatives and suppose that $$ f(x) \leq 0 \quad \text { and } \quad f^{\prime \prime}(x) \geq 0 \quad \text { for all } x $$ Prove that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is constant. (Hint: Observe that \(f^{\prime}: \mathbb{R} \rightarrow \mathbb{R}\) is increasing.)