1. Find the derivative of √(2x - 3) by using the definition method. 2. Evaluate the limit, lim (sin(ax) + bx) / (ax + sin(bx)) as x approaches 0. 3. Show that the rectangle of larg... 1. Find the derivative of √(2x - 3) by using the definition method. 2. Evaluate the limit, lim (sin(ax) + bx) / (ax + sin(bx)) as x approaches 0. 3. Show that the rectangle of largest possible area for a given perimeter P is a square. 4. Solve the linear differential equation: dy/dx + 2tan(x)y = sin(x). 5. Using the trapezoidal rule, compute ∫(from 0 to 2)(2x² - 1)dx with 4 intervals; find the absolute error of approximation from its actual value. 6. Using L'Hospital's rule, evaluate lim (x → 0)(x - sin(x))/x². 7. Using integration, find the area of x² + y² = a². Read more

Steps to Solve

1. Find the Derivative Using the Definition
   To find the derivative of $f(x) = \sqrt{2x - 3}$ using the definition, we apply the limit definition of a derivative:
   $$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$
   Substituting in the function:
   $$ f'(x) = \lim_{h \to 0} \frac{\sqrt{2(x + h) - 3} - \sqrt{2x - 3}}{h} $$
   Then simplify the expression.

2. Evaluate the Limit
   We need to evaluate:
   $$ \lim_{x \to 0} \frac{\sin(ax) + bx}{ax + \sin(bx)} $$
   Substituting $x = 0$ gives an indeterminate form $\frac{0}{0}$. We can apply L'Hôpital's rule:
   Differentiate the numerator and denominator separately, then re-evaluate the limit.

3. Show Rectangle of Largest Area
   We need to find the dimensions of a rectangle with a given perimeter $P$ that maximizes the area $A$.
   Set the rectangle width as $x$ and height as $y$.
   Using the perimeter constraint:
   $$ P = 2x + 2y $$
   Express $y$ in terms of $x$:
   $$ y = \frac{P}{2} - x $$
   Now express the area $A$:
   $$ A = x \left(\frac{P}{2} - x\right) $$ Maximize this area using calculus by taking the derivative and setting it to zero.

4. Solve the Linear Differential Equation
   Given $$ \frac{dy}{dx} + 2\tan x \cdot y = \sin x $$, we can find the integrating factor:
   $$ \mu(x) = e^{\int 2\tan x , dx} $$
   Integrate and solve for $y$ using the method of integrating factors.

5. Using the Trapezoidal Rule
   To compute:
   $$ \int_0^2 (2x^2 - 1) , dx $$
   with 4 intervals, calculate the width of each interval:
   $$ h = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5 $$
   Evaluate the function at each interval and apply the trapezoidal rule formula to approximate the integral.

6. Evaluate Using L'Hôpital's Rule
   For:
   $$ \lim_{x \to 0} \frac{x - \sin x}{x^3} $$
   We find it is an indeterminate form $\frac{0}{0}$. So, apply L'Hôpital's Rule multiple times until you reach a solvable limit.

7. Find Area Using Integration
   To find the area under the curve described by:
   $$ x^2 + y^2 = a^2 $$
   (it describes a circle), set up the integral for area calculation. Use polar coordinates if necessary, and apply integration techniques.