IB Mathematics Analysis and Approaches Topic AHL…

Consider the limit $\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{\sin ^{2} x}$ .

(a) Show that the limit is of indeterminate form and l'Hôpital's rule once.

(b) l'Hôpital's rule twice more to evaluate the limit.

(e) Show that when $p > 1$ , $\displaystyle\int_1^{\infty} x^{-p}\, dx = \frac{1}{p-1}$ .

(f) why $\displaystyle\int_1^{\infty} x^{-p}\, dx$ diverges when $0 < p < 1$ .

(g) Verify the result from part 5 by evaluating $\displaystyle\int_1^{\infty} x^{-3}\, dx$ .

(h) The function $f(x) = e^{-x^2}$ arises frequently in mathematics and statistics. It does not have an antiderivative that can be expressed in terms of elementary functions, so $\displaystyle\int_1^{\infty} e^{-x^2}\, dx$ cannot be evaluated using standard integration techniques. However, it is still possible to whether this integral converges.

(i) Hence show that $e^{-x^2} \leq e^{-x}$ for all $x \geq 1$ .

(j) Show that $\displaystyle\int_1^{\infty} e^{-x}\, dx = e^{-1}$ .

(k) Using the results from parts 9 and 10, why $\displaystyle\int_1^{\infty} e^{-x^2}\, dx$ must converge to a finite value.

(l) The technique used in part 12, bounding an unknown integral by a known convergent one, can be applied more broadly. Consider the function

g(x) = \frac{1}{1+x^3}

By finding a suitable comparison function whose improper integral converges, show that converges. an upper bound for the value of this integral.

(e) For that solution, find the exact value of $\int_{0}^{\frac{\pi}{3}} y(x)\cos x\,dx$ .

Consider $\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}$ .

(a) Show that the limit is indeterminate and l'Hôpital’s rule twice.

(b) Evaluate the limit by applying l'Hôpital’s rule a third time.

This question investigates the convergence and divergence of improper integrals: integrals whose upper limit extends to infinity. Through specific examples, you will develop a general test for convergence and then apply it to integrals that cannot be evaluated directly.

Consider the improper integral $\displaystyle\int_1^{\infty} x^{-2}\, dx$ , which is defined as $\displaystyle\lim_{R \to \infty} \int_1^{R} x^{-2}\, dx$ .

(a) Find $\displaystyle\int_1^{R} x^{-2}\, dx$ , giving your answer in terms of $R$ .

(b) Hence show that $\displaystyle\int_1^{\infty} x^{-2}\, dx = 1$ .

Show that $\displaystyle\int_1^{\infty} x^{-1/2}\, dx$ does not converge to a finite value.

(d) The results above suggest that the convergence of $\displaystyle\int_1^{\infty} x^{-p}\, dx$ depends on the value of $p$ . In this section you will exactly when this integral converges.

For $p > 0$ and $p \neq 1$ , show that

\int_1^{R} x^{-p}\, dx = \frac{R^{1-p} - 1}{1-p}

On the interval $-\frac{\pi}{2} \lt x \lt \frac{\pi}{2}$ the function $y$ satisfies

$\frac{dy}{dx} - y\tan x = \sec x.$

(a) Find an integrating factor for this equation, and hence show that its general solution is $y = (x+C)\sec x$ , where $C$ is an arbitrary constant.

(d) For that solution, find the exact value of $y\left( \frac{\pi}{3} \right)$ .

(b) the value of $C$ for which $y$ tends to a finite limit as $x \to \frac{\pi}{2}^{-}$ .

IB Mathematics Analysis and Approaches AHL 5.13—limits and L’hopitals Questionbank

IB Mathematics Analysis and Approaches AHL 5.13—limits and L’hopitals Questionbank

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IB Mathematics Analysis and Approaches AHL 5.13—limits and L’hopitals Questionbank

IB Mathematics Analysis and Approaches AHL 5.13—limits and L’hopitals Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 5.13—limits and L’hopitals Questionbank

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