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MATH 2301-102 and -103 Fall 2017 Calculus I Sample Test Questions

The tests are cumulative and can include Pre-Calculus material as described in the MATH 2301 Calculus I Student Handbook. Here we give some sample questions, mostly from old tests. They are split up by the new sections covered on each test. In some cases part of the problem is deciding which method to use, so you may be able to do the problem using methods from earlier sections. Solutions to these problems are not posted because posting them would encourage you to settle for familiarity (the posted solution makes sense) rather than mastery (you are sure your own solution is correct). I recommend you work in this order:

1. Read the book and understand the explanations and examples.
2. Do the text homework and online homework, which have answers available.
3. Do these sample problems. If you did the first two steps well enough, then you should be confident in your solutions.

PreCalculus; Sections 1.3 and 1.4

1. Determine whether each of the following statements is True or False. Correct answers are worth \(+3\), incorrect answers are worth \(-2\), and no answer is worth \(+1\).
   1. \(\frac{A+B}{A+C}=\frac{B}{C}\)
   2. \(\frac{A+B}{C+D}=\frac{A}{C}+\frac{B}{D}\)
   3. \(x+x^{-1}=0\)
   4. \((x+y)^2 = y^2+x^2+2xy\)
   5. \(x^3 + y^3 = (x+y)(x^2 + xy + y^2)\)
   6. \(\frac{\sin(x^3)}{\sin(x^2)}=\frac{x^3}{x^2}=x\)
   7. \(\sin(x+y)=\sin(x)+\sin(y)\)
   8. \(\sqrt{x^2+y^2}=\sqrt{x^2}+\sqrt{y^2}=x+y\)
   9. \(f(x)=x^{-1}\;\Rightarrow\;f(x+h)=x^{-1}+h\)
   10. A sphere of radius \(r\) has volume \(4\pi r^3/3\)
   11. If \(\displaystyle \lim_{x\rightarrow 4}f(x)=5\) then \(f(4)=5\).
2. With \(f(x)=\tan x\), \(g(x)=\frac{x}{x-1}\), and \(h(x)=\sqrt[3]{x}\), compute \((f\circ g\circ h)(x)\).
3. Verify the identity \(\frac{1}{1-\cos(\theta)}+\frac{1}{1+\cos(\theta)} = 2 \csc^2(\theta)\).
4. Solve the following equation for \(x\): \(\log_3(x-4)+\log_3(x+4) = 2\).
5. Consider \(y=f(x)\) with graph:
   
   1. Find the equations of the lines that comprise the graph of \(f\).
   2. Draw the graph of \(g(x)=2f(-x)+1\). Mark and label the points corresponding to \(A\), \(B\), and \(C\).
6. Use the properties of exponents to simplify \[ \left(\frac{25}{4x^4y^5}\right)\left(\frac{5}{2x^3y^2}\right)^{-3}\,.\]
7. Write the equation of the line passing through the two points \((1,3)\) and \((3,4)\).
8. Use the properties of logarithms to write \(f(x) = 2 \ln(x - 3) + \log_e (y + 2) - \ln(z)\) as a single logarithm.
9. Sketch the graph of a single function \(f\) that:
   1. has domain \([-4,5]\)
   2. has \(f(2) = 1\)
   3. has \(\displaystyle \lim_{x\rightarrow 2}f(x)=4\)
   4. has \(\displaystyle \lim_{x\rightarrow 3^+}f(x)=-3\)
   5. has \(\displaystyle \lim_{x\rightarrow 3^-}f(x)=3\)
10. Simplify and cancel so that you can plug in the given value without dividing by 0. Then plug in the value.
    1. For \(x=2\), \( \frac{x^2+x-6}{x-2}=\)
    2. For \(x=4\), \( \frac{\sqrt{x}-2}{x-4}=\)
    3. For \(h=0\), \( \frac{(x+h)^2 -x^2}{h}=\)
    4. For \(h=0\), \( \frac{(x+h)^{-1} - x^{-1}}{h}=\)
    5. For \(h=0\) and \(f(x)=2x^2\), \(\frac{f(x+h) -f(x)}{h}=\)
11. Compute the following limits. If you use the squeeze theorem, then indicate the two functions that you are using to squeeze.
    1. \(\displaystyle\lim_{x\rightarrow 2}\frac{x-2}{x^2-5x+6}\)
    2. \(\displaystyle\lim_{x\rightarrow 1}\frac{\sqrt{x}-1}{x-1}\)
    3. \(\displaystyle\lim_{x\rightarrow 0}x^2 \cos(3/x)\)
    4. \(\displaystyle\lim_{h\rightarrow 0}\frac{x^2-(x-2h)^2}{h}\)
    5. \(\displaystyle\lim_{t\rightarrow 0}\frac{\frac{1}{5+t}-\frac{1}{5}}{t}\)
    6. For \(f(x)= (2x+1)^{-1}\), compute \( \displaystyle\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\)
12. 1. State the Squeeze Theorem. Identify what are its assumptions (hypotheses) and what are its conclusions.
    2. Use the Squeeze Theorem to evaluate \[\lim_{x\rightarrow 0} \left(x^2 \cos\left(\frac{1}{x}\right)+1\right)\,.\]

Sections 1.5, 1.6, 2.1, 2.2, and 2.3

1. Determine whether each of the following statements is True or False. Correct answers are worth \(+3\), incorrect answers are worth \(-2\), and no answer is worth \(+1\).
   1. If \(f'(a)\) exists then \(\displaystyle\lim_{t\rightarrow a}f(t)=f(a)\).
   2. If \(f\) is continuous at \(a\) then \(\displaystyle\lim_{t\rightarrow a^+}f(t)=f(a)\).
   3. If the velocity of an object is constant, then its acceleration is zero.
   4. The function \(f(x)=x^{2/3}\) is differentiable on \((-\infty,\infty)\).
2. Let \[ f(x)= \begin{cases} {\displaystyle\frac{2x^2-x-15}{x-3}} & \text{if \(x \lt 3\)}\\ kx-1 & \text{if \(x \ge 3\)} \end{cases}\, . \] Determine the value of \(k\) that will make the function \(f\) continous at 3, or explain why no value of \(k\) will work.
3. Sketch the graph of a function \(f\) for which:
   1. It has a removable discontinuity at \(x=1\),
   2. \(\lim_{x\rightarrow 2^-}f(x)=+\infty\),
   3. \(\lim_{x\rightarrow 2^+}f(x)=-1\),
   4. \(\lim_{x\rightarrow +\infty}f(x)=3\),
   5. \(f'(x)>0\) everywhere it exists.
4. 1. State the Intermediate Value Theorem. Identify what are its assumptions (hypotheses) and what are its conclusions.
   2. Use the Intermediate Value Theorem to show that the equation \(x^2 =\cos(x)\) has a solution.
5. Consider the function \[f(x)= \begin{cases} x^2 & \text{if $x\le -2$}\\ Ax & \text{if $x> -2$} \end{cases}\, , \] where \(A\) is some constant.
   1. Find \(\displaystyle\lim_{x\rightarrow -2^-}f(x)\). Is \(f\) continuous from the left at \(x=-2\)?
   2. What value of \(A\) would make \(f\) continuous at \(x=-2\)?
   3. Using the value of \(A\) that you just found, graph \(f\).
6. Compute the following limits. If you use the squeeze theorem, then indicate the two functions that you are using to squeeze.
   1. \(\displaystyle\lim_{x\rightarrow 2^+}\frac{x+2}{x^2-5x+6}\)
   2. \(\displaystyle\lim_{x\rightarrow -\infty}\frac{3x^3-4}{2x^3-2}\)
   3. \(\displaystyle\lim_{x\rightarrow \infty}\cos(1/x)\)
   4. \(\displaystyle\lim_{x\rightarrow \infty} (x-x^2)\)
   5. \(\displaystyle\lim_{x\rightarrow \infty} \sin(\pi/x)\)
   6. \(\displaystyle\lim_{x\rightarrow -\infty}\frac{3x^3-4}{1+20x^3-2x}=\)
   7. \(\displaystyle\lim_{x\rightarrow \infty} (x-\sin(x))\)
7. Let \(f(x)=-x^2+3\).
   1. State the definition of the derivative as a limit.
   2. Using this definition, compute \(f'(x)\).
   3. Find the equation for the tangent line at \(x=2\).
   4. Graph \(f(x)\) and the tangent line.
8. Find the limit. What derivative does this limit represent? \[ \lim_{h\rightarrow 0} \frac{(x+h)^3-x^3}{h} \]
9. Find values for \(m\) and \(b\) so that \(\displaystyle f(x)= \begin{cases} x^2 & \text{if \(x\le -2\)}\\ mx+b & \text{if \(x> -2\)} \end{cases}\) is differentiable at \(x=-2\).
10. The graph of a function \(f\) is given in each part below. On the same axes, sketch the graph of \(f'\).
    
11. Sketch the graph of a function \(g\) for which \(g(0)=g'(0)=0\), \(g'(-1)=-1\), \(g'(1)=3\), and \(g'(2)=1\).
12. Compute the following derivatives:
    1. \(\displaystyle f(x)=2+x+\frac{3}{x}-\sqrt{x}-5x^7+x^{3/4} \Rightarrow f'(x)=\)
    2. \(\displaystyle\frac{d}{dx}\left[3\sin(x) + \cot(x) - \sin(7)\right]=\)
    3. \(\displaystyle y=\frac{x^3+x}{x} \Rightarrow \frac{dy}{dx}=\)
13. Complete the definitions:
    1. Definition: A function \(f\) is continuous at a number \(a\) if
    2. Definition: A function \(f\) is differentiable at \(a\) if
    Give an example of a function that is one but not the other.

Sections 2.4, 2.5, 2.6, 2.7, and 2.8

1. Compute the following derivatives:
   1. \(\displaystyle D_x \left[(9\cos(x)+x^8+x^5+3)\sin(x)\right]=\)
   2. \(\displaystyle y=\frac{x^3+5x}{\sin(x)} \Rightarrow \frac{dy}{dx}=\)
   3. \(\displaystyle D_x \left[\frac{\sin(x)\cos(x)}{x^3+x}\right]=\)
   4. \(\displaystyle D_x \left[(x^9+x^8+x^5+3)(1+2x^2+9x^3-4x^4)\right]=\)
   5. \(\displaystyle \frac{d}{dx} \left[(x^9+2x^{1/3}+x^5+3)^4\right]=\)
   6. \(\displaystyle y=(3+x^4)^8 x^3 \Rightarrow \frac{dy}{dx}=\)
   7. \(\displaystyle \frac{d}{dx} \left[ 5\tan(x^2 \sin(x^3+7x))\right]=\)
   8. \(\displaystyle D_x \left[\left( (x^9+x^8+x^5+3)(1+2x^2+x^3-4x^4)+1\right)^9\right]=\)
2. Use implicit differentiation to find an equation for the tangent line to the curve defined by \( \displaystyle x^2+4xy+y^2=13 \) at the point \((1,2)\).
3. If \(y=4x^3+x\) and \(\displaystyle\frac{dx}{dt}=5\), find \(\displaystyle\frac{dy}{dt}\) when \(x=2\).
4. A kite 100 feet above the ground moves horizontally at a speed of 8 feet per second. At what rate is the angle between the string and the horizontal decreasing when 200 feet of string has been let out?
5. At noon, ship \(A\) is 100 km directly west of ship \(B\). Ship \(A\) is sailing south at 35 km/hour and ship \(B\) is sailing north at 25 km/hour. How fast is the distance between the ships changing at 4:00pm? (Do not simplify your answer.)
6. The radius of a spherical cell is observed to decrease at a rate of \(2\) units/second when that radius is \(30\) units long. How fast is the volume of the cell decreasing at that point?
7. A trough is \(10 \mathrm{m}\) long and its ends have the shape of isosceles triangles that are \(3 \mathrm{m}\) across at the top and have a height of \(1 \mathrm{m}\). The trough is being filled with water at a rate of \(12\mathrm{m}^3/\mathrm{min}\). Draw and label a diagram illustrating this scenario. How fast is the water level rising when it is \(0.5\mathrm{m}\) deep?
8. The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. Use differentials to estimate the maximum relative error in the calculated area of the disk.
9. Use a linear approximation (or differentials) to estimate \(\displaystyle (8.03)^{2/3}\).

Sections 3.2, 3.3, 3.5, 3.6, and 3.7

1. Determine whether each of the following statements is True or False. Correct answers are worth \(+3\), incorrect answers are worth \(-2\), and no answer is worth \(+1\).
   1. If \(f\) is a one-to-one continuous function, then \(f^{-1}\) is a one-to-one continuous function.
   2. If \(f\) is a one-to-one differentiable function, then \(f^{-1}\) is a one-to-one continuous function.
2. The function \( f(x)=-7+ \sqrt[7]{4x-5}\) is one-to-one on its domain.
   1. Find a formula for its inverse, \(f^{-1}(x)\).
   2. Verify your formula is correct by computing and simplifying \(f\circ f^{-1}(x)\).
3. Let \(f(x)=7x-3\) and \(g(x)=\frac{x+3}{7}\).
   1. Compute \((g\circ f)(x)\).
   2. Compute \((f\circ g)(x)\).
   3. Are \(f\) and \(g\) inverses of each other?
4. State L'Hopital's Rule. Identify its assumptions (hypotheses) and its conclusions.
5. Compute the following limits. If you use the Squeeze theorem or L'Hopital's rule, then say so.
   1. \(\displaystyle\lim_{h\rightarrow 0}\frac{(3+h)^2-3^2}{h}=\)
   2. \(\displaystyle\lim_{x\rightarrow \infty} \frac{2}{x}\sin(3x)=\)
   3. \(\displaystyle\lim_{x\rightarrow \infty} \frac{2}{x}e^{3x}=\)
   4. \(\displaystyle\lim_{x\rightarrow \infty}\frac{e^{3x}}{x^3}=\)
   5. \(\displaystyle\lim_{x\rightarrow 1}\frac{\ln (x)}{\sin (\pi x)}=\)
   6. \(\displaystyle \lim_{x\rightarrow \infty} \ln(5x^9)=\)
   7. \(\displaystyle\lim_{x\rightarrow \infty} \arctan(e^x)=\)
   8. \(\displaystyle\lim_{x\rightarrow -\infty}\frac{3e^{3x}-4}{2e^{3x}-2}\)
6. Compute the following derivatives:
   1. \(\displaystyle \frac{d}{dx} \left[ x^5 \right]=\)
   2. \(\displaystyle \frac{d}{dx} \left[ 5^x \right]=\)
   3. \(\displaystyle \frac{d}{dx} \left[ 5^5 \right]=\)
   4. \(\displaystyle \frac{d}{dx} \left[ x^{1/5} \right]=\)
   5. \(\displaystyle \frac{d}{dx} \left[ x^{-5} \right]=\)
   6. \(\displaystyle \frac{d}{dx} \left[ \log_5(x) \right]=\)
   7. \(\displaystyle y=\frac{x^x \sin(2x) (x^5-7x)^6}{(\sqrt{x^9+1}) 3^x}\Rightarrow \frac{dy}{dx}=\)
   8. \(\displaystyle f(x)=\tan(x)+ \arctan(x) + \ln(x) + \log_4(x) +e^x \Rightarrow f'(x)=\)
   9. \(\displaystyle f(x)=\big({x^4+2x}\big)\big( {\log_3(\arctan(7x))}\big) \Rightarrow f'(x)=\)
   10. \(\displaystyle f(x)=x^x (\sin(2x))^{\cos(3x)} \Rightarrow f'(x)=\)
   11. \(\displaystyle f(x)=\tan(x)+ 2\arctan(x) + 3\ln(x) + \log_4(x) +5e^x \Rightarrow f'(x)=\)
   12. \(\displaystyle f(x)=\sin(1) +2^x + x^3 + x^{1/4} +5 \Rightarrow f'(x)=\)
   13. \(\displaystyle \frac{d}{dx} \left[ \frac{\big({3x-\cos(x)}\big)({x^2+2})}{x^4+5}\right]=\)
   14. \(\displaystyle \frac{d}{dx} \left[\left( (x^2+e^x)(x^3-4x^4)+7x\right)^9\right]=\)
   15. \(\displaystyle f(x)=\sinh(\cosh(x)) \Rightarrow f'(x)=\)
7. Use logarithmic differentiation to compute the derivative of \[\displaystyle y=\frac{x^x \sin(2x)}{3^x\sqrt{x^9+1}}\,.\]
8. Prove that \(\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\).
9. For the function \(\displaystyle f(x)=x^5-x^3+2x \), find \((f^{-1})'(2)\).
10. Find an equation of the tangent line to the graph of \(y = 3 \arccos(x/2)\) at \((1, \pi)\).
11. A ladder 10 ft long is leaning against a vertical wall. It starts slipping, such that the bottom of the ladder slides away from the base of the wall at a speed of \(2 \mathrm{ft}/\mathrm{s}\). Draw and label a diagram illustrating this scenario. How fast is the angle between the ladder and the wall changing when the bottom of the ladder is 6 ft from the base of the wall?

Sections 4.1, 4.2, 4.3, and 4.4 Show/ Hide

1. Determine whether each of the following statements is True or False.
   1. A local maximum of a function \(f(x)\) can only occur at a point where \(f'(x)=0\). Hint See Fermat's Theorem in section 4.1.
   2. If \(f\) is a continous function on the interval \([a,b]\) then \(f\) attains an absolute maximum at some point \(c\) in \((a,b)\). Hint See the Extreme Value Theorem in section 4.1.
   3. If \(f''(a)=0\) then \(f\) has an inflection point at \(x=a\). Hint See the definition of an inflection point in section 4.3.
   4. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(f+g\) is an increasing function on \(I\). Hint See the definition of increasing in section 1.1, think about its relationship to the derivative discussed in section 4.3, and try some simple test functions like \(f(x)=x\) and \(g(x)=x\).
   5. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(f-g\) is an increasing function on \(I\). Hint See the definition of increasing in section 1.1, think about its relationship to the derivative discussed in section 4.3, and try some simple test functions like \(f(x)=x\) and \(g(x)=x\).
   6. If \(f\) and \(g\) are increasing functions on an interval \(I\) then \(fg\) is an increasing function on \(I\). Hint See the definition of increasing in section 1.1, think about its relationship to the derivative discussed in section 4.3, and try some simple test functions like \(f(x)=x\) and \(g(x)=x\).
   7. If \(f\) and \(g\) are positive increasing functions on an interval \(I\) then \(fg\) is an increasing function on \(I\). Hint See the definition of increasing in section 1.1, think about its relationship to the derivative discussed in section 4.3, and try some simple test functions.
   [There are more True/False questions, and answers to some of these, in the chapter review.]
2. Find the absolute maximum and minimum values of \(f\) on the given interval. [For example functions and their solutions, see section 4.1 #37-49 odd.]
3. 1. State Rolle's Theorem using the template below.
      • If ...
      • then ...
   2. Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. [For example functions and their solutions, see section 4.2 #1, 3.]
4. 1. State the Mean Value Theorem (MVT) using the template below.
      • If ...
      • then ...
   2. Verify that the function satisfies each of the hypotheses of the MVT on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the MVT. [For example functions and their solutions, see section 4.2 #9, 11.]
5. Sketch the graph of a single function that has all of the following properties:
   1. ....
   2. ....
   [For example property lists and their solutions, see section 4.3 #17-21 odd.]
6. 1. Find the intervals where \(f\) is increasing, and the intervals where it is decreasing.
   2. Find the local maximum and minimum values
   3. Find the intervals where \(f\) is concave up, and the intervals where it is concave down.
   4. Find the inflection points
   5. Use this information to sketch the graph.
   [For example functions and their solutions, see section 4.3 #25, 27, 29, 33, 35.]
7. 1. Find the domain.
   2. Find the intercepts.
   3. Determine any symmetries.
   4. Find any asymptotes.
   5. Find the intervals on which \(f\) is increasing or decreasing.
   6. Find the local maximum and minimum values of \(f\).
   7. Find the intervals of concavity and the inflection points.
   8. Use the information above to sketch the graph.
   [For example functions and their solutions, see section 4.4 #5-17 odd, 21, 27, 31, 33, 37, 39, 41. For some of these finding the \(x\)-intercept is too hard and can be omitted.]
8. Let \(f\) be a continuous function with \(f(0)=3\), \(f(2)=6\), \(f'(x)=0\) for \(0 < x < 1\), and \(f'(x) < 2\) for \(1 < x < 2\). Sketch such a function or explain why it is impossible. Hint Break into intervals and use the Mean Value Theorem.

Sections 4.5, 4.6, and 4.7 Show/ Hide

1. (Optimization word problem) [For example problems with solutions (for odd numbers), see section 4.5 #5-17 odd, 16, 21, 25, 27, 29, 33, 36, 43, 45, 47, 54, 56, 58]
2. Use Newton's method with the specified initial approximation \(x_1\) to find \(x_2\), the second approximation to the root of the given equation. Leave the answer as a fraction. [For example problems, see section 4.7 #6-8. Check your work with the sage tool.]
3. Find the function \(f\) given its derivative and value at certain points. [For example problems with solutions, see section 4.7 #17-31 odd.]
4. (Antiderivative word problem) [For example problems with solutions (for odd numbers), see section 4.7 #43, 46, 47, 51, 53, 54.]

Sections 5.1, 5.2, and 5.3 Show/ Hide

1. True or False: If \(f(x) \gt 0\) and \(f'(x) \gt 0\) then the Midpoint Rule will overestimate the integral.
2. Estimate the area under the graph of \(f(x)\) on the given interval using four rectangles and midpoints, and then using left endpoints. [For example functions and solutions, see section 5.1 #3-7 odd. There will not be too many intervals and you do not need to total the sums.]
3. (Estimates of area-like things from data) [For example functions and solutions, see section 5.1 #9, 11, 13. You do not need to total the sums.]
4. Evaluate the integral by interpreting it in terms of areas. [For example functions and solutions, see section 5.2 #31, 33, 35.]
5. Evaluate the integrals. [For example functions and solutions, see section 5.3 #1-25 odd, 43-47 odd.]
6. (Word problem using definite integrals) [For example functions and solutions, see section 5.3 #63, 65, 67.]
7. Suppose \(f\) and \(g\) are differentiable functions with the following properties: \[ \begin{array}{lll} f(0)=2 &f(1)=0 &f(2)=1 \\ g(0)=1 &g(1)=2 &g(2)=0 \\ \int_0^1 f(x)dx=\pi &\int_1^2 f(x)dx=\pi^3 &\int_2^3 f(x)dx=\pi^5 \\ \int_0^1 g(x)dx=\sqrt{2} &\int_1^2 g(x)dx= \sqrt{3} &\int_2^3 g(x)dx=\sqrt{5}\\ f'(0)=e &f'(1)=e^3 &f'(2)=e^5 \\ g'(0)=\sqrt{7} &g'(1)=\sqrt{11} &g'(2)=\sqrt{13} \end{array} \] Evaluate the following. If one cannot be evaluated with the given information, write "NOT ENOUGH INFORMATION."
   1. \(\displaystyle\int_0^1 f(r)dr\)
   2. \(\displaystyle\int_0^3 f(x)dx\)
   3. \(\displaystyle\int_3^2 g(x)dx\)
   4. \(\displaystyle\int_1^2 (5f(x)+g(x))dx\)
   5. \(\displaystyle\int_0^1 f(x)g(x)dx\)
   6. \(\displaystyle\int_0^{14} f(x)dx - \int_2^{14} f(x)dx\)
   7. \(\displaystyle \int_0^2 f'(r) \; dr\)
   8. \(\displaystyle \int_6^6 f''(x) \; dx\)
   9. \(\displaystyle\displaystyle \lim_{x\rightarrow 1}\frac{f(x)}{g(x)-2}\)
   10. \(\displaystyle\displaystyle \lim_{h \to 0}\frac{f(2+h)-1}{h}\)

Last modified: Tue Oct 31 14:50:23 UTC 2017