Exercises

Practice consolidates “differentiating upper-limit integrals” and related techniques.

练习题

练习 1

Let $F(x) = \int_0^x t^2 \, dt$. Find $F'(x)$.

练习 2

Let $F(x) = \int_1^x \dfrac{1}{t^2} \, dt$. Find $F'(x)$.

练习 3

Let $F(x) = \int_0^{x^3} \cos t \, dt$. Find $F'(x)$.

练习 4

Let $f(x) = \int_0^x e^{-t^2} \cos t\,dt$. Find $f'(0)$ and $f''(0)$, and decide whether $x=0$ is an extremum of $f(x)$.

练习 5

改编自 2022 考研数学一第 1 题

Given $\lim\limits_{x\to1} \dfrac{f(x)}{\ln x}=1$ with $f(x) = \int_0^x e^{t^2} \sin t\,dt$, find $f'(1)$.

练习 6

改编自 2023 考研数学一第 3 题

Let $y=f(x)$ be given parametrically by $\begin{cases}x=2t+|t|\\y=|t|\sin t\end{cases}$ and $f(x) = \int_0^x e^{t^2} \sin t\,dt$. Find $f'(0)$.

练习 7

改编自 2024 考研数学一第 1 题

Given $f(x)=\int_0^x e^{\cos t} dt$, find $f'(x)$ and $f''(x)$.

练习 8

改编自 2025 考研数学一第 1 题

Given $f(x) = \int_0^x e^{t^2} \sin t\,dt$, find $f'(x)$ and $f''(x)$.

练习 9

改编自 2022 考研数学一第 17 题

Given $y'= -\dfrac{1}{2\sqrt{x}}y + 2 + \sqrt{x}$ with $y(1)=3$, and $f(x) = \int_0^x e^{t^2} \sin t\,dt$, find $f'(1)$.

练习 10

改编自 2023 考研数学一第 14 题

Let continuous $f(x)$ satisfy $f(x+2)-f(x)=x$, $\int_0^2 f(x)dx=0$, and $F(x) = \int_0^x e^{t^2} \sin t\,dt$. Find $F'(0)$ and $F''(0)$.

总结

本文出现的符号

中英对照