Math 242: Calculus I

Midterm Exam: Review Guide

Part I, Computations -- Thursday, 18 October 2018

1. Compute the limits, provided they exist.

a.)	\(\displaystyle \lim_{x \to 0} \frac{x^2 - 2x + 1}{x - 1} \)	b.)	\(\displaystyle \lim_{x \to 1} \frac{x^2 - 2x + 1}{x - 1} \)	c.)	\(\displaystyle \lim_{\theta \to \tfrac{\pi}{2}^-} \tan\theta \)
d.)	\(\displaystyle \lim_{x \to 0} \dfrac{\sin(4x)}{x} \)	e.)	\(\displaystyle \lim_{y \to \tfrac{\pi}{4}} \ln(\tan(y))\)	f.)	\(\displaystyle \lim_{u \to \frac{\pi}{12}} \sin(u)\cos(u) \)

2. Compute the derivatives of the functions.

a.)	\( f(x) = 3x^2 + 9x - 5\)	b.)	\( g(x) = \dfrac{\sqrt{x} - \sqrt[3]{x}}{\sqrt[5]{x}}\)	c.)	\(h(x) = x^2\sin(x + 1) \)
d.)	\(v(\theta) = \tan\theta\cos\theta \)	e.)	\( y = \tan(\cos(x)) \)	f.)	\( K(p) = \sec^2(p) \)
g.)	\( P(T) = \dfrac{nRT}{V} \)	h.)	\( q(x) = -\dfrac{x}{\sqrt{1 + x^2}} \)	i.)	\( d(x) = \sqrt{1 - (x-1)^2} \)
j.)	\( f(r) = (2r - 1)^3(r + 1)^2 \)	k.)	\( T(t) = \dfrac{2 - \tan(t)}{\cos^2(t)} \)	l.)	\( u(y) = \csc(\sqrt{y^3 - 2y^2 + y}) \)

3. Find \(\dfrac{dy}{dx}\):    \(x^2 + 6xy + y^2 = \sin(xy) \)

4. Find equations of the tangent and normal lines to the curve \(y = \sqrt{1 - x^2} \) at the point \(P\left(\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2}\right)\).

5. Find all points on the curve \(y = x^2 - 2x + 1\) where the tangent line is parallel to the line \(2x - 4y = 1\).

Part II, Proofs and Applications -- Friday, 19 October 2018

6. You wish to prove that \(\displaystyle \lim_{x \to 2} 12x - 4 = 20\). If you fix \(\varepsilon > 0\), what should you set \(\delta\) equal to in order to finish the proof?

7. Show that the equation \(x^4 - 6x^2 = -5\) has a real root in the interval \((0,2)\).

8. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the interval. Then find the values of \(c\) that are guaranteed by the theorem. \[ f(x) = x^3 - 3x^2 + x - 1,\ \ \ 0 \leq x \leq 2 \]

9. Use the limit definition of derivative to compute \(\tfrac{dy}{dx}\) for the given functions. You must use the limit definition to receive credit on the exam.

	a.)	\(y = \sqrt{x}\)
	b.)	\(y = \dfrac{1}{x}\)
	c.)	\(y = x^5\)

10. Consider the piecewise function \[F(x) = \begin{cases} x^2 + 2x -1 & x<0 \\ k & x = 0 \\ \cos(x - \pi) & x > 0 \end{cases}.\] Is it possible for \(F\) to be continuous at \(x = 0\)? If so, what must \(k\) be equal to? Explain.

11. Find the Taylor polynomial \(T_5\) for the function \(f(x) = \sin(x)\) at \(x = 0\).

12. Find the Taylor polynomial \(T_3\) for the function \(f(x) = \sin(x)\) at \(x = \tfrac{\pi}{4}\).

13. Use a linear approximation or differentials to approximate the values of \(\sqrt{25.1}\) and \(\sqrt{24.99}\).

14. A 29 ft ladder is sliding down a wall at a constant rate of 4 inches per minute. Find the rate at which the base of the ladder is sliding away from wall when the base of the ladder is 20 ft from the base of the wall.

15. Find all relevant information about the graphs of the functions, then sketch the graph.

	a.)	\(y = x^3 - 3x^2 - 9x + 27\)
	b.)	\(y = \dfrac{\sqrt{x^2 - 1}}{x - 1}\)
	c.)	\(y = \dfrac{x^3}{x^2 + 3x + 2}\)

16. You wish to construct a rectangular box without a top such that the surface area of the box is 10 square inches. What are the dimensions of the box that maximize the volume?

17. You wish to construct an aluminum bucket in the shape of a right circular cylinder with a volume of 1 L. If the bucket does not have a top, find the dimensions of the cylinder that minimize the surface area.

18. Suppose \(f\) is continous on \([1,5]\), \(f(1) = 2\), and \(3 \leq f'(x) \leq 5\) for all \(x\) in the interval \((1,5)\). What are the largest and smallest possible values of \(f(5)\)?

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