Taylor Series and Power Series Exercise 1 : Expansion of a Function in a Power Series Expand the following functions at $x_0 = 0$ in a series up to the first four terms: $f(x) = \sqrt{1 - x}$ $f(t) = \sin(\omega t + \pi)$ $f(x) = \ln[(1 + x)^5]$ $f(x) = \cos x$ $f(x) = \cosh x$ Exercise 2 : Interval of Convergence of a Power Series Obtain the radius of convergence of the following series: ...
Differential Equations Exercise 1 : Classification — Linear vs Nonlinear Which of the following are linear first- and second-order DEs with constant coefficients? $y’ + x^2 y = 2x$ $5y’’ - 2y’ - 4x = 3y$ $y^4 + 2y’’ + 3y’ = 0$ $\sin x, y’’ - y = 0$ $y’’ - x^5 = 2$ $2y’’ - y’ + \frac{3}{2}y = 0$ Exercise 2 : Homogeneity and Order Which of the following are homogeneous and non-homogeneous DEs and what is the order in each case? ...
Differential Calculus Exercise 1 : Sequences and Limits of Sequences Calculate the limiting value of the following sequences for $n \to \infty$. $a_n = \frac{\sqrt{n}}{n}$ $a_n = \frac{5 + n}{2n}$ $a_n = \left(-\frac{1}{4}\right)^n - 1$ $a_n = \frac{2}{n} + 1$ $a_n = \frac{n^3 + 1}{2n^3 + n^2 + n}$ $a_n = 2 + 2^{-n}$ $a_n = \frac{n^2 - 1}{(n + 1)^2} + 5$ Exercise 2 : Limits of Functions Calculate the following limits: ...
Applications of Integration Exercise 1 : Area Bounded by Parabola Calculate the area bounded by the positive branch of the parabola $y^2 = 25x$, the $x$-axis and the ordinates where $x = 0$ and $x = 36$. Exercise 2 : Area Between a Symmetric Curve and the x-axis Calculate the area bounded by the positive branch of the curve $y^2 = (7 - x)(5 + x)$, the $x$-axis and the ordinates where $x = -5$ and $x = 1$. ...
Integral Calculus Exercise 1 : Primitives Find the primitives of the following functions and the value of the constant: $f(x) = 3x$ given $F(1) = 2$ $f(x) = 2x + 3$ given $F(1) = 0$ Exercise 2 : Definite Integrals (Cosine) Evaluate the following definite integrals: $\int_{0}^{\pi/2} 3 \cos x , dx$ $\int_{-\pi/2}^{\pi/2} 3 \cos x , dx$ $\int_{0}^{\pi} 3 \cos x , dx$ Exercise 3 : Absolute Areas Obtain the absolute values of the areas corresponding to the following integrals: ...
Derivatives Exercise 1 Using the definition, prove that if $f(x)=1/x$, then $f’(a)=-1/a^{2}$ for $a\neq 0$. Show that the tangent line to $f$ at $(a, 1/a)$ intersects $f$ only at $(a, 1/a)$. Exercise 2 Using the definition, prove that if $f(x)=1/x^{2}$, then $f’(a)=-2/a^{3}$ for $a\neq 0$. Show that the tangent line at $(a, 1/a^{2})$ intersects $f$ at exactly one other point, on the opposite side of the y-axis. Exercise 3 Prove that if $f(x)=\sqrt{x}$, then $f’(a)=1/(2\sqrt{a})$ for $a>0$. (Hint: Rationalize the difference quotient.) ...
Fundamental Theorem of Calculus Exercise 1 : Derivatives of Integral Functions Find $F’(x)$ for each function: $F(x) = \int_a^{x^3} \sin^3 t dt$ $F(x) = \int_3^x \frac{1}{1 + \sin^6 t + t^2} \int_1^t \sin^3 u du dt$ $F(x) = \int_{15}^x \int_8^y \frac{1}{1 + t^2 + \sin^2 t} dt dy$ $F(x) = \int_x^b \frac{1}{1 + t^2 + \sin^2 t} dt$ $F(x) = \int_a^b \frac{x}{1 + t^2 + \sin^2 t} dt$ $F(x) = \sin \left( \int_0^x \sin \left( \int_0^y \sin^3 t dt \right) dy \right)$ ...
Inverse Functions Exercise 1 Find $ f^{-1} $ for each function $ f $ $ f(x) = x^3 + 1 $ $ f(x) = (x - 1)^3 $ $ f(x) = \begin{cases} x & \text{rational} \ -x & \text{irrational} \end{cases} $ $ f(x) = \begin{cases} -x^2 & x \geq 0 \ 1 - x^3 & x < 0 \end{cases} $ $ f(x) = \begin{cases} x & x \neq a_i \ a_{i+1} & x = a_i \ (i < n) \ a_1 & x = a_n \end{cases} $ $ f(x) = x + \lfloor x \rfloor $ $ f(0.a_1a_2a_3\ldots) = 0.a_2a_1a_3\ldots $ $ f(x) = \frac{x}{1 - x^2}, \ -1 < x < 1 $ Exercise 2 Describe $ f^{-1} $’s graph when $ f $ is: ...
Riemann Sums Exercise 1 : Approximation of Integral Products Let \(f,g\) be continuous on \([a,b]\). For any partition \(P = \{t_0,\ldots,t_n\}\) of \([a,b]\), choose points \(x_i, u_i \in [t_{i-1},t_i]\). Show that sums of the form \[\sum_{i=1}^n f(x_i)g(u_i)(t_i-t_{i-1})\] can be made arbitrarily close to \(\int_a^b fg\) by choosing sufficiently fine partitions \(P\). Exercise 2 : Approximation of Composite Functions Let \(f,g\) be continuous and nonnegative on \([a,b]\). Show that for sufficiently fine partitions \(P\), sums ...
Convexity and Concavity Exercise 1 Show that a function $ f $ is convex on an interval if and only if for all $ x, y $ in the interval and $ 0 < t < 1 $, we have: $$ f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y). $$Exercise 2 Let $ f $ and $ g $ be convex functions with $ f $ increasing. ...