MATH 141 Spring 2015 Review Exam
 Find domain and range of the following functions:
\begin{align} f(x) &= \sqrt{x}  2 \\ f(x) &= \sqrt{49x^2} \\ f(x) &= \frac{5x+1}{\sqrt{x^2+5x+6}} \end{align}
 Compute the following limits:
\begin{align} &\lim_{x\to 1^+} \frac{x1}{x^21} \\ &\lim_{x\to 2} \frac{x^2+x6}{x2} \\ &\lim_{x\to 0^} \frac{\sqrt{x+1}1}{x} \\ &\lim_{x\to 3} \frac{19}{3x} \\ &\lim_{x\to 1} \frac{2x^2+2}{x^2+4x4} \\ &\lim_{x\to 1} \bigg( \frac{2+4x}{4+11x^2+3x^4} \bigg)^3 \end{align}
 Solve for \( x \) in the equation \( \log_4 (x+1) + \log_{16}(x+1) = \log_4 8 \),
 Simplify \( \log_7 2401 \).
 Expand the following expression:
\begin{equation} \log_3 \bigg( \frac{9x^{11}}{\sqrt{y}} \bigg) \end{equation}
 Find the value of the constant \( n \) that will make the function \( f(x) \) continuous everywhere.
\begin{equation} f(x) = \begin{cases} \frac{x^21}{x1} &\text{ if } x<1 \\ nx2 &\text{ if } x\geq 1 \end{cases} \end{equation}
 Find \( f \circ g \) and its domain, if
 \( f(x) = \sqrt{x}, g(x) = x4 \).
 \( f(x) = 3x^24x+7, g(x) = \sqrt{x \tfrac{2}{x}} \).
 \( f(x) = \dfrac{1}{x+2}, g(x) = \dfrac{3}{x3} \).
 \( f(x) = \dfrac{x}{x1}, g(x) = \sqrt{x} \).
 Simplify the following expressions
 \( 8^{4x} 8^{2000x} \).
 \( e^{12x^2} e^{73x} \).
 \( e^{x^2} e^{8x} e \).
 Find an exponential function of the form \( f(x) = Ca^x \), if we know that its graph contains the points \( (3,3) \) and \( (9,30) \).
 What is the effect of the following operations on the graph of \( f(x) \)?
 \( 4f(x) \).
 \( f(6x) \).
 \( 5 f(x3)4\).

Find the derivative of the following functions
\begin{align} f(x) &= 230x^{300} + 15x^{18} + 5 \\ f(x) &= \frac{488}{233} x^{265/488} \\ f(x) &= 2\sin x + 6\cos x + 9 \\ f(x) &= \sin x \cos x \\ f(x) &= 12\tan x + 3x^2  \cos x \\ f(x) &= \sin^2 x + \cos^2 x \\ f(x) &= e^{8x} \\ f(x) &= 8^x \\ f(x) &= 10^{2x^2} \end{align} 
Use the chain rule to find the derivative of the following function
\begin{equation} f(x) = \sqrt{ \big( 3x^5 + \sin(e^x)  400 \big)^3 } \end{equation} 
Use the quotient rule to obtain the derivative of
\begin{equation} f(x) = \frac{x^3}{x^2 + 6} \end{equation}  Find the tangent line to the graph of the function \( f(x) = x^3  x^2 + 60 \) at \( x=2 \).
 Find the normal line to the graph of function \( f(x) = 4x^221x+4 \) at \( x=3 \).

Use logarithmic differentiation to obtain the derivative of the following functions
\begin{align} f(x) &= \big( \cos 8x \big)^x \\ f(x) &= \big( \tan x \big)^{9/x}\\ \end{align}  Find all \( x \)values in the interval \( [0,2\pi] \), where the tangent line to the graph of \( f(x)= \sin^2 x  \cos^2 x \) is horizontal.

Use implicit differentiation to obtain the derivative \( y’ \) (as a function of \( x\)), if \( x \) and \( y \) are related by the following equations:
\begin{align} e^{x+y} &= y^3 \\ x^2 + y^2 &= 25 \end{align}  Sketch the graph of the following functions. You must find zeros, domain, range, vertical and horizontal asymptotes, intervals of increase/decrease, and intervals of concavity. Indicate also local extrema and inflection points.
\begin{align} f(x) &= 4x^2+24x+32 \\ f(x) &= \frac{x}{x24} \end{align}

Find the critical points of \( f(x) = x^{1/3}  x^{2/3} \).
 Compute the following limits:
\begin{align} &\lim_{x\to \infty} e^{6x}x^{1/2} \\ &\lim_{x\to \infty} \frac{23x14x^3+7}{4x^2+7} \end{align}
 Find all critical points of the function \( f(x) = \dfrac{x4}{x^22x+8} \),
 Find the absolute extrema of the following functions, in the indicated intervals:
\begin{align} f(t) &= 2\cos t + \sin 2t && [0, \tfrac{\pi}{2}] \\ f(x) &= xe^{x^2/8} && [1, 4] \end{align}

A farmer wants to fence an area of 24 million square feet and then divide it in half with a fence parallel to one of the sides of the rectangular area. What should the length of the two sides be to minimize fencing?

A ladder 12 ft long is resting against a vertical wall. If the ladder is sliding away from the wall at 0.5 ft/sec, how quickly is the angle between the ground and ladder changing when the ladder is 4 ft from the base of the wall?

A square box with an open top has a volume of 10 cubic meters. Material for the base costs $5 per square meter. Material for the sides costs $2 per square meter. What is the cost of materials for the cheapest such container?
 A piece of wire 10 m long is cut into 2 pieces. One piece is built into a square and the other is built into an equilateral triangle.
 How much wire should be used for the square in order to maximize the total area?
 How much wire should be used for the square to minimize the total area?

A flashlight lying on the ground shines on a building 15 yards away as a 5yard tall man walks towards the same building at 2 yd/s. How fast is the length of the shadow decreasing when he is exactly half way between the flashlight and the building?

A man is 6 feet tall and walking away from a streetlight that is 15 feet tall at a rate of 5 ft/sec. At what rate is his shadow increasing when he is 20 feet from the streetlight?

A cylindrical tank is filled with water. The tank stands upright and has a radius of 10 meters. How fast does the height of the water drop when it is drained at 15 cubic meters per second?

At 3 pm, ship A is 120 km west of ship B. Ship A is sailing east at 25 km/hour and ship B is sailing north at 35 km/h. How fast is the distance between the ships changing at 7 pm?

Gravel is being dumped from a conveyor belt at a rate of 20 cubic feet per minute and forms the shape of a cone whose base, diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

The height of a rectangular box is 10 inches. Its length increases at a rate of 2 in/sec and its width decreases at the rate of 4 in/sec. When the length is 8 inches and the width is 6 inches, the volume of the box is changing at the rate of?

A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km/h and run 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows.)

Find two numbers whose difference is 100 and whose product is a minimum

Find the next three terms, and the general term of the sequence that starts as \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \dotsc \).
 Evaluate the following limits:
\begin{align} \lim_{n\to \infty} & \sum_{i=1}^n \bigg[ \big(\frac{i}{n} \big)^2 + \frac{i}{n} \bigg] \\ \lim_{n \to \infty} & \sum_{i=1}^n \frac{6}{n} \bigg[ \big(\frac{2i}{n} \big)^2 + 5 \big(\frac{2i}{n}\big) \bigg] \end{align}
 Compute the following sums:
\begin{align} \sum_{k=1}^{200} 7k^3  3k^2 + 15 &&& \sum_{n=1}^{1000} (4n+n)(2n^2+7) \end{align}
 Compute the following integrals:
\begin{align} &\int 3x^3+2x+5+\frac{6}{x^2}\, dx && \int \frac{1}{4}x^2+\frac{2}{3}x+\frac{19}{7}\, dx && \int_1^e \frac{1}{4x}\, dx \\ &\int \sqrt{18x} + 89\sin x\, dx && \int \sec^2 x + \cos x  \sin x\, dx &&\int \big( x\tfrac{1}{x} \big)\, dx \\ &\int e^{5x+8}\, dx && \int (2^x + 3^x e^x)\, dx && \int_1^3 \big( \frac{1}{x}  \frac{1}{x^2} \big)\, dx \end{align}
 Find the net area of the following functions, in the indicated intervals:
\begin{align} f(x) &= x^4 && [1, 3] \\ f(x) &= \sin x && [0, 2\pi] \end{align}