Calculus Solutions Ex# 3.6

Q 1-2. Evaluaté the given limit without using L’Hopital rule, and then check that your answer is correct using L’Hopital rule. : 1. (a) $$ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{2}+2 x-8} $$ $$ \begin{aligned} & \lim_{x \rightarrow 2} \frac{(x-2)(x+2)}{x+4 x-2 x-8}=\lim_{x \rightarrow 2} \frac{(x-2)(x+2)}{x(x+4)-2(x+4)} \\ \\ \Rightarrow \quad & \lim_{x \rightarrow 2} \frac{(x-2)(x+2)}{(x+4)(x-2)}=\lim_{x \rightarrow 2} … Read more

Calculus Solutions EX # 3.5

Summary: Local linear appronimation of $f$ at $x_{0}$ : Local linear appronimation of $f$ at $x_{0}$ : $$ f(x) \approx f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)—–(1) $$ Let $\Delta x=x-x_{0}$ then $$ f\left(x_{0}+\Delta x\right) \approx f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right) \Delta x——(2) $$ Differentials: $$ \begin{aligned} & d y=f^{\prime}(x) d x ———-(5)\\ & f^{\prime}(x)=\lim {\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} \\ \\ & … Read more

Calculus Solutions Ex # 3.4

Q1. Equation $y=3 x+5$ (a) Given that $\frac{d x}{d t}=2$, find $\frac{d y}{d t}$ when $x=1$ (b) Given that’ $\frac{d y}{d t}=-1$, find $\frac{d x}{d t}$ when $x=0$. . Solution: (a) $$ y=3 x+5 $$ Differentiating w.r.t’ $t$ ‘ $$ \begin{aligned} & \frac{d y}{d t}=\frac{3 d x}{d t} \\ \\ & \left.\frac{d y}{d t}\right|_{x=1}=\left.3 \frac{d … Read more

Calculus Solutions Ex# 3.3

Formulas: Derivatives of exponential and inverse trigonometric functions functions. $$ \begin{aligned} & \text { (1) }\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)} \text { or for } x=f(y) , \frac{d y}{d x}=\frac {1}{dx/dy} \\ \\ & \text { (2) } \frac{d}{d x}\left[b^{x}\right]=b^{x} \ln b \\ \\ & \text { (3) } \frac{d}{d x}\left[e^{x}\right]=e^{x} \end{aligned} $$ If $u$ is a differentiable function … Read more

Calculus Solutions Ex# 3.2

Formulas: Derivatives of logarithmic functions.   $$ \begin{aligned} & \text { (1) } \frac{d}{d x}[\ln x]=\frac{1}{x}, \quad x>0 \\ \\ & \text { (2) } \frac{d}{d x}\left[\log _{b}^{x}\right]=\frac{1}{x \ln b}, x>0 \end{aligned} $$ Generalized derivative formulas. $$ \begin{aligned} & \text { (1) } \frac{d}{d x}[\ln u]=\frac{1}{u} \frac{d u}{d x} \\ \\ & \text { (2) … Read more

Calculus Solutions Ex#3.1

Q.01 (a) Find $\frac{d y}{d x}$ by differentiating implicitly. Solution: $$ y=x+x y-2 x^{3}=2—(1) $$ Differentiating w.r.t $x$ $$ \begin{aligned} & \frac{d}{d x}\left[x+x y-2 x^{3}\right]=\frac{d}{d x}[2] \\ \\ & \frac{d}{d x}[x]+\frac{d}{d x}[x y]-2 \frac{d}{d x}\left[x^{3}\right]=0 \\ \\ & 1+x \frac{d y}{d x}+y \frac{d}{d x}[x]-2\left(3 x^{3-1}\right)=0 \\ \\ & 1+\frac{x d y}{d x}+y-6 x^{2}=0 \\ \\ … Read more