Lesson 55 — Higher-Order Derivatives

Let $f(x) = x^3 - 2x^2 + x - 5$. Calculate $f'(x)$ and $f''(x)$.

Let $f(x) = x^5 - 3x^2 + x + 2$. Calculate $f''(x)$.

Let $f(x) = \sin x$. Calculate $f''(x)$.

Let $f(x) = \cos(2x)$. Calculate $f''(x)$.

Let $f(x) = \ln x$. Calculate $f''(x)$.

Let $f(x) = xe^x$. Calculate $f''(x)$.

Let $f(x) = x^2 \ln x$. Calculate $f''(x)$.

Let $f(x) = x^4 - 4x^3 + 1$. Calculate $f'''(x)$.

Let $f(x) = \dfrac{1}{1 + x^2}$. Calculate $f''(0)$.

Let $f(x) = \sqrt{x}$. Calculate $f''(x)$.

Let $f(x) = \cos(2x)$. Calculate $f^{(4)}(x)$.

Let $f(x) = x^4$. Calculate $f^{(5)}(x)$.

Let $f(x) = e^{2x}$. Determine $f^{(n)}(x)$ for all $n \geq 1$.

Determine $(\sin x)^{(100)}$.

Let $f(x) = \dfrac{1}{x}$. Determine the general formula $f^{(n)}(x)$.

For $f(x) = x^4 - 4x^3 + 1$, determine the inflection points and intervals of concavity.

For $f(x) = x^3 - 3x^2 + 2$, determine the intervals of concavity and the inflection point.

For $f(x) = e^{-x^2}$, calculate $f''(0)$.

For $f(x) = x^5 - 5x^4$, determine the inflection points.

If $f''(c) = 0$, can we conclude that $c$ is an inflection point of $f$?

If $f'(c) = 0$ and $f''(c) > 0$, what can be concluded about $c$?

Determine the concavity of $f(x) = e^x$ throughout its domain.

Analyze the concavity of $f(x) = x^3$ and identify the inflection point.

For $f(x) = x^4 - 6x^2$, determine the intervals of concavity and the inflection points.

Explain why $(\sin x)^{(4)} = \sin x$ and why $(e^x)^{(n)} = e^x$ for all $n \geq 0$.

Derive the formula for $(fg)''$ from the product rule, and identify the analogy with Newton's binomial theorem.

Let $f(x) = (1 + x)^{10}$. Calculate $f^{(10)}(0)$.

Position of a particle: $s(t) = 4t^3 - t^4$ (meters, $t$ in seconds). Calculate $v(1)$, $a(1)$, and $j(1)$, and interpret $j(1) = 0$.

Pendulum: $\theta(t) = A\cos(\omega t)$. Calculate $\ddot{\theta}$ and verify that $\ddot{\theta} + \omega^2\theta = 0$.

Production cost: $C(q) = q^3 - 6q^2 + 15q$ (R$thousand). Calculate$C''(q)$ and interpret the inflection point as "minimum marginal cost".

Vehicle position: $s(t) = 10t^3 - 30t^2 + 5$ (meters). Calculate $v(t)$, $a(t)$, $j(t)$ and determine when acceleration is zero.

Projectile height: $h(t) = -4{,}9t^2 + v_0 t + h_0$. Calculate $h''(t)$ and identify its physical meaning.

In a mechanical system, the potential energy $U(\theta)$ has a critical point at $\theta_0$. What does $U''(\theta_0) > 0$ versus $U''(\theta_0) < 0$ imply about the stability of the equilibrium?

Using the first three derivatives of $f(x) = e^x$ at $a = 0$, write the Taylor polynomial $T_2(x)$ and estimate the error for $x = 0{,}1$.

Write the Taylor polynomial of degree 2 for $f(x) = \cos x$ around $a = 0$ and verify for $x = 0{,}1$.

Calculate $f^{(n)}(x)$ for $f(x) = \ln(1+x)$ and write the Taylor polynomial $T_n(x)$ around $a = 0$.

For $f(x) = x^x$ ($x > 0$), calculate $f''(x)$ using logarithmic differentiation.

State Leibniz's formula $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(k)} g^{(n-k)}$ and describe the structure of the induction argument that proves it.

Proof. Let $f$ be twice differentiable on $[0, 1]$, with $f(0) = f(1) = 0$ and $f' \not\equiv 0$. Does there exist $c \in (0, 1)$ with $f''(c) = 0$? Justify.

Proof. Prove that if $f$ is twice differentiable and $f''(x) \geq 0$ on $(a, b)$, then $f$ is convex on $(a, b)$.