MATH 141 Spring 2015 Review Exam
- Find domain and range of the following functions:
\begin{align} f(x) &= -\sqrt{x} - 2 \\ f(x) &= -\sqrt{49-x^2} \\ f(x) &= \frac{5x+1}{\sqrt{x^2+5x+6}} \end{align}
- Compute the following limits:
\begin{align} &\lim_{x\to 1^+} \frac{x-1}{x^2-1} \\ &\lim_{x\to 2} \frac{x^2+x-6}{x-2} \\ &\lim_{x\to 0^-} \frac{\sqrt{x+1}-1}{x} \\ &\lim_{x\to 3} -\frac{19}{3-x} \\ &\lim_{x\to -1} \frac{2x^2+2}{x^2+4x-4} \\ &\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^2-1}{x-1} &\text{ if } x<1 \\ nx-2 &\text{ if } x\geq 1 \end{cases} \end{equation}
- Find \( f \circ g \) and its domain, if
- \( f(x) = \sqrt{x}, g(x) = x-4 \).
- \( f(x) = 3x^2-4x+7, g(x) = \sqrt{x- \tfrac{2}{x}} \).
- \( f(x) = \dfrac{1}{x+2}, g(x) = \dfrac{3}{x-3} \).
- \( f(x) = \dfrac{x}{x-1}, g(x) = \sqrt{x} \).
- 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 \( f(x) \)?
- \( 4f(x) \).
- \( -f(6x) \).
- \( 5 f(x-3)-4\).
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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 \( f(x) = 4x^2-21x+4 \) at \( x=3 \).
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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.
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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}{x-24} \end{align}
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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{23x-14x^3+7}{4x^2+7} \end{align}
- Find all critical points of the function \( f(x) = \dfrac{x-4}{x^2-2x+8} \),
- 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}
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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?
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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?
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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?
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A flashlight lying on the ground shines on a building 15 yards away as a 5-yard 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?
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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?
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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?
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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?
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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?
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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?
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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.)
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Find two numbers whose difference is 100 and whose product is a minimum
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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}