Solved Problems

Category: Calculus

Solved Problems in Calculus

Problem 2-8: Integration

Find $\displaystyle\int \! \frac{e^x}{1+e^x}\, dx$ .

Solution

Let $u=1+e^x$ , therefore $du=e^x dx \to dx=\frac{1}{u-1}du$ . Hence

$\displaystyle\int \! \frac{e^x}{1+e^x}\, dx = \displaystyle\int \! \frac{u-1}{u} \frac{1}{u-1}\,du = \displaystyle\int \! \frac{1}{u} \,du=\ln (u)=\ln(1+e^x)$

April 10, 2010
Problem 2-7: Evaluatin Definite Integrals

Evaluate

a) $\displaystyle\int_{-1}^1 \! x^2 \, dx$
b) $\displaystyle\int_{-2}^1 \! \sin(x) \, dx$
c) $\displaystyle\int_{-1}^2 \! 2x^3+x-1 \, dx$
d) $\displaystyle\int_{-2}^2 \! |x| \, dx$

Solution
a) $\displaystyle\int_{-1}^1 \! x^2 \, dx=\frac{1}{3}x^3\Bigr|^1_{-1}=\frac{1}{3}(1)^3-\frac{1}{3}(-1)^3=\frac{2}{3}$
(more…)

April 10, 2010
Problem 2-6: Odd and Even Functions

Which one of the following functions are even or odd or neither?
a) $\sin(x)$
b) $\cos(x)$
c) $\sin(x)\cos(x)$
d) $x \sin(x)$
e) $x^2$
f) $\sin(x) + \cos(x)$

Recall that a function is said to be even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$ .

Solution
a) Odd $\sin(-x)=-\sin(x)$

b) Even $\cos(-x)=\cos(x)$
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April 10, 2010
Problem 2-5: Evaluating Indeterminate Form

Calculate $\displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}$ .

Solution
$\frac{\sqrt{x+2}-2}{x-2}=\frac{0}{0}$
$\displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}= \displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}\times \frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$

$=\displaystyle\lim_{x\to 2}\frac{x+2-4}{(x-2) \times (\sqrt{x+2}+2)}$
$=\displaystyle\lim_{x\to 2}\frac{x-2}{(x-2) \times (\sqrt{x+2}+2)}$

$=\displaystyle\lim_{x\to 2}\frac{1}{(\sqrt{x+2}+2)}=\frac{1}{4}$

April 9, 2010
Problem 2-4: Computing Limits of a Rational Function

Compute
a) $\displaystyle\lim_{x\to 3}\frac{x^2+2x-3}{x+3}$
b) $\displaystyle\lim_{x\to -3}\frac{x^2+2x-3}{x+3}$

Solution

a) $\displaystyle\lim_{x\to 3}\frac{x^2+2x-3}{x+3}=\frac{3^2+2\times 3-3}{3+3}=2$
(more…)

April 9, 2010
Problem 2-3: Solving an Inequality

Solve the inequality $|2-3x|<3$ .

Solution

$|2-3x|<3$
$2-3x<3$ or $-(2-3x)<3$

$-3x<1$ or $3x<5$

$-\frac{1}{3} < x<\frac{5}{3}$

April 9, 2010
Problem 2-2: Evaluating Derivative of Functions and the Tangent Lines

Find the derivative of $f(x)$ and the equation of the tangent line at $x_0=-1$ .

a) $f(x)=x^2$
b) $f(x)=x^3+x+1$
c) $f(x)=\frac{1}{x}$

Solution
The equation of the tangent line at ${x}_{0}$ is $y = f'(x_0) (x-x_0) + f(x_0)$ .
a) $f(x)=x^2, f'(x)=2x$
$y = f'(x_0) (x-x_0) + f(x_0) = 2x_0 (x-x_0)+x_0^2 = 2x_0 x-x_0^2$
$y = -2x-1$ .

b) $f(x)=x^3+x+1$

$f'(x)=3x^2+1$
$y=f'(x_0) (x-x_0) + f(x_0) =(3x_0^2+1) (x-x_0)+x_0^3+x_0+1=(3x_0^2+1)x-2x_0^3+1$
$y=4x+3$ .

c) $f(x)=\frac{1}{x}=x^{-1}$
$f'(x)=-x^{-2}$ $f'(x)=-\frac{1}{x^2}$
$y=f'(x_0) (x-x_0) + f(x_0) =-\frac{1}{x_0^2}(x-x_0)+\frac{1}{x_0}=-\frac{1}{x_0^2} x +\frac{2}{x_0}$
$y = -x-2$

April 7, 2010
Problem 2-1: Linearity of Functions

Which one(s) of the following functions is linear?

a) $y=x+1$ .
b) $y=2x+1$ .
c) $y=1$ .
d) $y=x^2+x+1$ .
e) $y=x^1+1$ .
f) $y=\frac{1}{x}$ .
g) $y=\sin(x)$ .

h) $y=\sqrt{x}$ .
i) $y=x+\frac{1}{1-x}$ .

j) $5x+2y-1=0$ .
k) $y=x+c^2$ . ( $c$ is an arbitrary constant).

Solution
a) Linear
b) Linear
c) Linear
d) Nonlinear
e) Linear
f) Nonlinear

g) Nonlinear
h) Nonlinear
i) Nonlinear
j) Linear

k) Linear

April 5, 2010