Final Practice

These are a collection of problems you can use to practice for the final for the Chapter 6 content. To see the answers for Chapter 6, go to this page.

Remember, look at the Exam II practice problems in order to get practice on the Chapter 5 content.

P-level

  1. Compute the derivative of the following functions.

    1. \(f(x)=10^{x}\)
    2. \(f(x)=\arctan(x)\)
    3. \(\displaystyle f(x)=\arctan(x-1)3^x+\mathrm{e}^{x^2}+\frac{x}{\cos(x)+1}\)
    4. \(\displaystyle f(x)=\ln(x-2)+\tan(x-1)+\frac{5x^2-3^4}{\sin(x)}\)

  2. Determine the following indefinite integrals.

    1. \(\displaystyle\int 4x\,\mathrm{d}x\)
    2. \(\displaystyle\int x^2\,\mathrm{d}x\)
    3. \(\displaystyle\int \sec^2(5x)\,\mathrm{d}x\)
    4. \(\displaystyle\int 3^{5x}\,\mathrm{d}x\)
    5. \(\displaystyle\int \left(\frac{4}{\sqrt{1-x^2}}-5\cos(x)+\pi^2\right)\,\mathrm{d}x\)
    6. \(\displaystyle \int \left(\frac{\pi^2}{x^2+1}-3\sec(x)\tan(x)+\mathrm{e}^{4x}\right)\,\mathrm{d}x\)

  3. Take \(f\) to be a continuous and differentiable function such that

    • \(f^\prime\) is positive on \((-\infty,3)\),
    • \(f^\prime\) is negative on \((3,\infty)\),
    • \(f^\prime\) is zero on the set \(\{3\}\),
    • \(f^{\prime\prime}\) is positive on \((-\infty,0)\cup(4,\infty)\),
    • \(f^{\prime\prime}\) is negative on \((0,4)\),
    • \(f^{\prime\prime}\) is zero on the set \(\{0,4\}.\)

    Determine on which intervals \(f\) is increasing and decreasing.

  4. Take \(f\) to be a continuous and differentiable function such that

    • \(f^\prime\) is positive on \((-\infty,3)\);
    • \(f^\prime\) is negative on \((3,\infty)\);
    • \(f^\prime\) is zero on the set \(\{3\}\);
    • \(f^{\prime\prime}\) is positive on \((-\infty,0)\cup(4,\infty)\);
    • \(f^{\prime\prime}\) is negative on \((0,4)\);
    • \(f^{\prime\prime}\) is zero on the set \(\{0,4\}\);

    Determine on which intervals \(f\) is convex and concave.

  5. Take \(f\) to be a twice differentiable function such that \(f^\prime\) is

    • positive on \((-\infty, 4)\cup(5,7)\cup(7,\infty)\);
    • negative on \((4,5)\);
    • zero on the set \(\{4,5,7\}\);
    • increasing on \((-\infty,2)\), and \((4.5,8)\);
    • decreasing on \((2,4.5)\), and \((8,\infty)\).

    Determine on which intervals \(f^{\prime\prime}\) is positive and negative.

  6. Take \(f\) be a function such that an antiderivative of \(f\) is \(F(x)=\arctan(x-1)\). Determine \(\displaystyle\int_0^1f(x)\,\mathrm{d}x.\)

  7. Take \(F\) to be the function defined on \([\sqrt{2},\infty)\) given by \(F(x)=\displaystyle\int_2^{x^2}\sin(t)\cos(t)\,\mathrm{d}t\). Determine the derivative of \(F\).

  8. Take \(f\) to be the function whose sketch is given below. Determine all intervals on which \(f^\prime\) is positive and negative.

  1. Take \(f\) to be a function defined on \([-5,5]\) whose graph is given below. Determine all intervals at which \(f\) is differentiable on.

  1. The graph of a function \(f\) is given below on the domain \([-3,5]\). Determine an interval \(I\) so that all hypotheses of Rolle’s Theorem are met. Determine an interval \(J\) so that one of the hypotheses of Rolle’s Theorem is not met.

  1. Take \(f\) to be a differentiable function such that \(|f^\prime(x)|\leq 2\) on \([-3,4].\) Determine a modulus of continuity for \(f\).

  2. Take \(f\) to be a continuous function on \([-4,4]\) with the following data: \[f(-4)=2,\quad f(-3)=4,\quad f(-2)=1,\quad f(-1)=-2,\quad f(0)=4,\quad f(1)=2,\quad f(2)=2,\quad f(3)=3,\quad\text{and}\quad f(4)=-3.\] Identify the smallest intervals guaranteed to a zero of \(f\) and explain why.

  3. Take \(f\) and \(g\) to be differentiable functions with \[f(1)=4, \quad g(1)=3, \quad f^\prime(1)=\tfrac{1}{2},\quad f^\prime(3)=2\quad \text{and} \quad g^\prime(1)=3.\] Determine the following

    1. \((f+3g)^\prime(1)\)
    2. \((fg)^\prime(1)\)
    3. \(\left(\frac{f}{g}\right)^\prime(1)\)
    4. \(\left(f\circ g\right)^\prime(1)\)
    5. \((\pow_2\circ f+\exp\circ g)^\prime(1)\)

C-level

  1. Take \(f\) to be the function given by \(f(x)=\sin\left(x+\frac{\pi}{2}\right)\). Determine the local linear approximation of \(f\) at \(x=-\frac{\pi}{2}.\)

  2. Take \(f\) to be the function given by \(f(x)=\sqrt{3x+7}.\) Determine the local linear approximation of \(f\) at \(x=3.\)

  3. Take \(f\) to be the function given by \(f(x)=x(2^{x})\). Determine the line that is tangent to \(f\) at \(x=1\).

  4. Take \(g\) to be a differentiable function. Compute the derivative of the following functions:

    1. \(\displaystyle f(x)=\log_2(x)\ln(x)+\sin(x-1)+\frac{5x^2-4^x}{\sec(x)}+g(x^2)\)
    2. \(\displaystyle f(x)=\arccos(2x-4)\cos(x)+4^{x^2+3x}+\sqrt{g(x)-1}\)

  5. Take \(y=y(x)\) to be implicitly defined by each equation given below. Determine \(\frac{\mathrm{d}y}{\mathrm{d}x}.\)

    1. \(y+\tan(x)=\cos(y-1)-\csc(x)\)
    2. \(x+x\tan(y)=\cos(y-1)-\cot(x)\)

  6. Determine the following indefinite integrals:

    1. \(\displaystyle\int \left(\frac{2x}{\sqrt{x^2+1}}+3(5x-4)^3\right)\,\mathrm{d}x\)
    2. \(\displaystyle\int \left(\frac{2x}{x^2+1}+x\mathrm{e}^{-x^2}\right)\,\mathrm{d}x\)
    3. \(\displaystyle\int \frac{\mathrm{e}^x}{1+\mathrm{e}^{2x}}\,\mathrm{d}x\)
    4. \(\displaystyle\int \frac{40x^3+5x+1}{10x^4+\frac{5}{2}x^2+x+20}\,\mathrm{d}x\)
    5. \(\displaystyle\int \frac{\mathrm{e}^x+\ln(2)2^x}{(\mathrm{e}^x+2^x)^2}\,\mathrm{d}x\)

  7. Determine the following definite integrals:

    1. \(\displaystyle\int^4_0 \left(2f(x)+x-4+2\cos(2x)\right)\,\mathrm{d}x\) where \(\displaystyle\int_0^4f(x)\,\mathrm{d}x=-2\)
    2. \(\displaystyle\int^3_1 \left(3f(x)+4-\frac{4}{x}\right)\,\mathrm{d}x\) where \(\displaystyle\int_1^3f(x)\,\mathrm{d}x=2\)

  8. Take \(f\) to be a differentiable function such that \(f^\prime(x)=(x-1)(x+3)^2.\) Determine all maximal intervals on which \(f\) increasing and decreasing. Determine all local maximum and local minimum.

  9. Take \(f\) to be a differentiable function such that \(f^\prime(x)=x(x+3)^2(x-5)^6.\) Determine all maximal intervals on which \(f\) increasing and decreasing. Determine all local maximum and local minimum.

  10. Take \(f\) to be a differentiable function such that \(f^\prime(x)=x\mathrm{e}^x-10x.\) Determine all maximal intervals on which \(f\) increasing and decreasing. Determine all local maximum and local minimum.

  11. A spherical balloon is being inflated with air at a rate so its volume is increasing at a rate of \(125\) centimeters cubed per second. Determine the rate of change of the radius of the balloon when the diameter is \(30\) centimeters.

  12. The position, \(p\), of a particle at time \(t\) is given below as a sketch on the interval \([-4,2]\).

    1. Determine at which times the particle is at rest.
    2. Determine at which times the particle moves in the positive direction.
    3. Determine at which times the particle moves in the negative direction.

  1. Take \(f\) to be a differentiable function such that the sketch of \(f^{\prime}\) is given below. List the extreme values of \(f\) and determine if they are local maxima or local minima.

  1. Take \(f\) to be a function defined on \([-2,5]\), twice differentiable function on \([-2,0)\cup(0,2.5)\cup(2.5,5]\), and the graph of \(f^\prime\) on \([-2,5]\) is given below.
    1. Determine all intervals on which \(f\) is increasing and decreasing.
    2. Determine all intervals on which \(f\) is convex and concave.
    3. Determine all local maximum and minimum of \(f\).
    4. Determine all inflection points of \(f\).

  1. Take \(c\) to be a path defined on \(\mathbb{R}\) such that it is at \((1,2)\) at time \(0\) and its velocity \(v\) is given by \[v(t)=\left\langle t^2+1,2\cos(t)+4t\sqrt{t^2+1} \right\rangle.\] Determine a formula for the path \(c\).

B-level

  1. A spherical balloon is being filled with air at a rate of four cubic inches per second. Determine how fast the surface area changes when the radius of the balloon is \(12\) inches.

  2. Apply Newton’s method to a suitable quadratic polynomial to estimate \(\sqrt{19}\) with an initial guess of \(x_0=4\). Stop at the third iteration of the method and draw a picture that graphically illustrates what you are doing.

  3. Take \(f\) to be the function given by \(f(x)=\frac{1}{3}x^3+\frac{1}{2}x^2-6x+1\). Determine all local maximum, local minimum, global maximum and global minimum of \(f\) on the interval \([-5,4]\) by using the first derivative test and Fermat’s theorem.

  4. Take \(f\) to be the function given by \(f(x)=\frac{1}{3}x^3+2x^2+4x+3.\) Determine all local maximum, local minimum, global maximum and global minimum of \(f\) on the interval \([-5,4]\) by using the first derivative test and Fermat’s theorem.

  5. Take \(f\) to be the function given by \(f(x)=|x|(-x^{2}+4 x-2)-3.\) Determine all local maximum, local minimum, global maximum and global minimum of \(f\) on the interval \([-1,3]\) by using the first derivative test and Fermat’s theorem.

  6. Use L’Hopital’s rule to calculate these limits:

    1. \(\displaystyle\lim_{x\to 0^+}x^2\log_5(x)\)
    2. \(\displaystyle\lim_{x\to 1^+}(x-1)^{5(x-1)}\)
    3. \(\displaystyle\lim_{x\to \infty}\left(\sqrt{4x^2+9x+1}-2x\right)\)
    4. \(\displaystyle\lim_{x\to \infty}\left(1+\frac{1}{3x}\right)^{\tfrac{x}{5}}\)

  7. Take \(c\) to be a path defined on \(\mathbb{R}\) such that it is at \((1,2)\) at time \(0\), its velocity is \(\langle 2,4\rangle\) at time \(0\), and its acceleration is given by \[a(t)=\left\langle t^2+t+1,\sin(t)+\mathrm{e}^{2t} \right\rangle.\] Determine a formula for the path \(c\).

A-level

  1. Take \(f\) to be a function that is defined and differentiable on the interval \([-1,4],\) and \(|f'(x)|\leq \frac{1}{2}\) on \([-1,4]\). Use the mean value theorem to sketch the smallest region in the plane that is guaranteed to contain \(f\) given these points in \(f.\)
    1. \((-1,2)\)
    2. \((2,2)\)
    3. \((4,2)\)
    4. \((-1,2)\), \((2,2)\), and \((4,2)\)
    5. In each of these four cases, find the smallest interval that is guaranteed to contain the range of \(f.\)
  2. Sketch a differentiable function \(f\) on \(\mathbb{R}\) that has all the following properties. Label any local min, local max, and inflection points.
    • \(f^{\prime}\) is positive on \((-\infty,3)\),
    • \(f^{\prime}\) is negative on \((3,\infty)\),
    • \(f^{\prime}\) is zero on the set \(\{3\}\),
    • \(f^{\prime\prime}\) is positive on \((-\infty,0)\cup(4,\infty)\),
    • \(f^{\prime\prime}\) is negative on \((0,4)\),
    • \(f^{\prime\prime}\) is zero on the set \(\{0,4\}.\)
  3. Take \(f\) to be the function given by \[f(x)=\begin{cases}3x^2-8&\text{if }0\leq x<2\\7&\text{if }2\leq x\leq 10.\end{cases}\] Take \(g\) to be the function whose graph is given below.
    1. Calculate \(\displaystyle\lim_{x\to 9^-}(f\circ g)(x)\)
    2. Calculate \(\displaystyle\lim_{x\to 2^-}(g\circ f)(x)\)
    3. Calculate \((f+g)^\prime(1)\)
    4. Calculate \((fg)^\prime(4)\)
    5. Calculate \(\displaystyle\int_0^{10} (f(x)+g(x))\,\mathrm{d}x\)
    6. Calculate \(\displaystyle\int_2^6 (f(x-2)+g(x-2))\,\mathrm{d}x\)
    7. Calculate \(\displaystyle\lim_{x\to 4}\frac{f(x-3)+5}{g(x)-2.5}\)

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