MAT 180: Exams


  • Final Exam Outline:
    • Solving polynomial and rational inequalities.
    • Simplifying algebraic expressions which involve factoring common factors with positive and negative exponents.
    • Identifying the domain of a given function.
    • Determine the difference quotient for polynomial, rational, and square root functions.
    • Sketching the graph of a function by shifting, reflecting, and/or stretching the known graph of a related function.
    • Graphing a piecewise defined function. Writing a function that involves absolute value as a piecewise defined function.
    • Finding and verifying the inverse of a given function.
    • Graphing a degree three or higher polynomial function using intercepts and end behavior.
    • Finding the (real or complex) roots of a polynomial equation using the Rational Root Test and synthetic division.
    • Graphing a rational function by locating intercepts and horizontal and vertical asymptotes.
    • Graphing exponential and logarithmic functions, especially base \(e.\)
    • Expanding or condensing logarithmic expressions.
    • Solving exponential or logarithmic equations, especially base \(e.\) Checking for extraneous roots.
    • A word problem involving solving exponential or logarithmic equations, especially base \(e.\) Checking for extraneous roots.
    • Given stated conditions, finding the constants in a general exponential function \(y = ae^{bx}.\) This may involve a word problem.
    • Given the value of a trigonometric function, find the exact value of another trigonometric function.
    • A word problem involving solving a right triangle.
    • Graphing a sine or cosine, function by finding the period, amplitude and shift. Graphing a tangent function by finding the period and shift.
    • Verify a trigonometric identity.
    • Simplify a trigonometric expression.
    • Solve a trigonometric equation.
    • Use the trigonometric addition formulas for sine and cosine.
    • Use the trigonometric double angle formulas for sine and cosine.
    • Find the exact value of an expression involving inverse trigonometric functions and trigonometric functions.
    • Use the Law of Sines and the Law of Cosines.

    Review Problems

  • Exam 3 Outline:
    • State the change-of-base formula
    • State the properties of logarithms: Power Property, Quotient Property, Product Property, Inverse Property
    • Apply the change-of-base formula
    • Given a graph(s) of a logarithmic and/or exponential function, answer questions related to the graph(s). For example, find the asymptote, find an equation of the function that coincides with the graph, find \(x\)-intercepts, find the \(y\)-intercept, etc…
    • Solve exponential equations of various types. There will be at least one involving factoring. For example, solve \(e^{2x} =-4e^x-5 .\)
    • Solve logarithmic equations of various types. There will be at least one requiring isolating and consolidating logarithms on one side. For example, solve \(\log(x+5) = \log(x-1) + \log(x+1).\)
    • Graph an exponential function. For example, graph \(f(x) = 3e^{4x-1} -10.\) Find and label the \(x\) and \(y\)-intercepts, and asymptotes (if any). Also include a table of “nice” values where to evaluate the exponential function and the corresponding values of the function.
    • Graph a logarithmic function. For example, graph \(f(x) = -2\ln (-5x+4) +9.\) Find and label the \(x\) and \(y\)-intercepts, and asymptotes (if any). Also include a table of “nice” values where to evaluate the logarithmic function and the corresponding values of the function.
    • Expand and condense logarithmic expressions.
    • Given values of logarithmic expressions, use the property of logarithms to find the exact value of a logarithmic expression. For example, if \(\log_2 u = 19\) and \(\log_2 v = -30\), find the exact value of \(\frac{\log_2 (u^2 v)}{\log_2 u – \log_2 v} + \log_2 v^3\)
    • Evaluate logarithms. For example \(\log_2 64, \ln e, \log_a 1, \) etc…
    • A problem involving finding a formula \(y = a b^{kx -c} + d\) that satisfies some given set of conditions. This may possibly be a word problem. For example, suppose that bacteria grow according to an exponential growth model. If the initial population of bacteria is 12131 and after 10 years the population is 1452345, find the exponential growth model for the total amount of bacteria after \(t\) years and use it to predict the population after 20 years.

    Exam 3

  • Exam 2 Outline:
    • You will be asked to state some of the formulas that correspond to the following named identities:
      • Pythagorean Identity
      • Power-reducing formulas for sine and cosine
      • Sum and difference formulas for sine and cosine
      • Double-angle formulas for sine and cosine
    • Verify an identity
    • Find the exact value of a trigonometric function for a nonstandard angle. For example, \(\cos \left( \frac{7 \pi}{12} \right).\)
    • Find the exact values of \(\sin 2u, \cos 2u, \) and \(\tan 2u\) given some information about one of the six basic trigonometric functions. For example, find the exact value of \(\sin 2u, \cos 2u, \) and \(\tan 2u\) given that \(\sin u = \frac{-13}{14}\) and \( \pi < u < \frac{3 \pi}{2}. \)
    • Find the exact value of either \(\cos(u\pm v), \sin(u \pm v),\) \(\tan(u \pm v), \sec(u \pm v),\) \( \csc(u \pm v), \cot(u \pm v)\) given some information about the six basic trigonometric functions. For example, find the exact value of \(\csc(u-v)\) given that \(\sin u = \frac{1}{3}\) and \(\cos v = \frac{3}{4}. \) (Both \(u\) and \(v\) are in Quadrant III.)
    • Solve basic trigonometric equations on a restricted interval. For example, find all solutions to (a) \(\tan x = -1\) on the interval \([0, 2\pi)\), (b) \(\cos x = 5\) on \((0, \pi]\), (c) \(\sin x = 1/2 \) on \([-\pi,0)\), etc…
    • Solve, not necessarily on a restricted domain, a multiple angle trigonometric equation. For example, solve \(\sin 7 x = \frac{1}{2}\).
    • Solve, not necessarily on a restricted domain, a trigonometric equation by factoring.
    • Solve, not necessarily on a restricted domain, a sum-of-angles trigonometric equation. For example, solve \(\cos(\pi-x) – \cos x + 1 = 0\).
    • Solve a trigonometric equation using inverse trigonometric functions. For example, solve \(\cos 5x = \frac{-2}{9}\)

    Exam 2

  • Exam 1 Outline:
    • Given a graph of a trigonometric function, you’ll be asked to answer questions related to the graph. For example, find the amplitude, phase shift, period, asymptotes, \(x\)-intercepts, \(y\)-intercept, find a formula whose graph is the one given, etc…
    • You will be asked to fill-in a blank unit circle with the common angles between \(0\) and \(2 \pi\) and the corresponding ordered pairs.
    • There will be a question requiring you to find the values of the six basic trigonometric functions given some information about the value of one of them. For example:
      Find the values of the six trigonometric functions of \(\theta\) given that \(\tan \theta = \frac{-15}{8}\) and \(\sin \theta >0\).
    • Graph a trigonometric function: This will either be sine, cosine, or tangent. You’ll need to provide a nice table of values. Your sketch will need to include \(x\)-intercepts, \(y\)-intercepts, two full periods, and labels for asymptotes (if any).
    • Graph a trigonometric function: This will either be cosecant, secant, or contangent. You’ll need to provide a nice table of values. Your sketch will need to include \(x\)-intercepts, \(y\)-intercepts, one full period, and labels for asymptotes (if any).
    • Simplify expressions involving composition of trigonometric and inverse trigonometric functions. For example, \(\csc \left[ \arctan \left(-\frac{5}{12} \right)\right]\).
    • One or two questions involving solving a trigonometric equation, on the entire real line or on a proper subinterval. For example, solve \(3\sin(5x+1) -2 = 1\) or solve \(3 \sin(5x+1) -2 = 1\) on \([0, 2\pi).\)
    • Graph \(f(x) = a\arctan x+ b \), where \(a\) and \(b\) are constants. For example, graph \(f(x) = 2 \arctan x + \frac{\pi}{4}.\)
    • Evaluate sine, cosine, tangent, secant, cosecant, cotangent, arcsin, arccos, and arctan. For example, quiz 1 and P1 of quiz 2

    Exam 1