10. Q10c
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Similar Question 1
<p>Analyse and sketch the function.</p><p><code class='latex inline'>\displaystyle f(x) = 2x^3 -3x^2 -3x + 2 </code></p>
Similar Question 2
<p>Sketch the graph of a function <code class='latex inline'>y = f(x)</code> that satisfies each set of conditions.</p><p><code class='latex inline'>\displaystyle \lim_{x\to -1^+}f(x) = 3, \lim_{x\to -1^-}f(x) = 2</code>, and <code class='latex inline'>f(-1) = 5</code></p>
Similar Question 3
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>\displaystyle h(x) = 3x^2 -27 </code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<ul> <li>i. Find the local maximum and minimum values.</li> <li>ii. Find the intervals of concavity and the inflection points.</li> <li>iii. Summarize all information on a table as shown in the lecture.</li> <li>iv. Sketch the graph.</li> </ul> <p><code class='latex inline'>f(x) = (x^2 - 1)^3</code></p>
<p>Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.</p><p>The domain of <code class='latex inline'>f</code> is <code class='latex inline'>\mathbb{R}</code>.</p><p><code class='latex inline'>\displaystyle \lim_{x\to \infty} f(x) = -\infty </code> and <code class='latex inline'>\displaystyle \lim_{x\to -\infty} f(x) = -\infty </code></p><p>The y-intercept is 2. The x-intercepts are -1, 3, and 5. There is a local extrema at x = 3.</p>
<p>For each function,</p><p>i) How does the function behave as <code class='latex inline'>x\to \infty</code>? Explain your reasoning.</p><p>ii) Does the function have any symmetry? Explain.</p><p>iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.</p><p>iv) Find the point(s) of inflection and make a table showing the intervals of concavity.</p><p><code class='latex inline'> \displaystyle f(x) = \frac{x}{x^2 + 1} </code></p>
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>\displaystyle f(x) = x^5 -2x^4 + 3 </code></p>
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>\displaystyle h(x) = 3x^2 -27 </code></p>
<p>Sketch each function.</p><p><code class='latex inline'>h(x)=x^5+20x^2+5</code></p>
<p>Sketch each function.</p><p><code class='latex inline'>k(x)=\displaystyle{\frac{1}{2}}x^4-2x^3</code></p>
<ul> <li>i. Find the local maximum and minimum values.</li> <li>ii. Find the intervals of concavity and the inflection points.</li> <li>iii. Summarize all information on a table as shown in the lecture.</li> <li>iv. Sketch the graph.</li> </ul> <p><code class='latex inline'> f(x) = 2x^3 - 3x^2 - 12x + 1 </code></p>
<ul> <li>i. Find the local maximum and minimum values.</li> <li>ii. Find the intervals of concavity and the inflection points.</li> <li>iii. Summarize all information on a table as shown in the lecture.</li> <li><p>iv. Sketch the graph.</p></li> <li><p> <code class='latex inline'>f(x) = x^4 + 4x^3</code></p></li> </ul>
<p>Sketch each function using suitable techniques.</p><p><code class='latex inline'>f(t)=\displaystyle{\frac{t^2-3t+2}{t-3}}</code></p>
<p>Sketch a graph of a function f that is differentiable on the interval <code class='latex inline'>-3\leq x\leq 4</code> and that satisfies the following conditions:</p><p><strong>a)</strong> The function <code class='latex inline'>f</code> is differentiable on <code class='latex inline'>-1 < x < 3</code> and increasing elsewhere on <code class='latex inline'>-3\leq x \leq 4 </code></p><p><strong>b)</strong> The largest value of <code class='latex inline'>f</code> is 6, and the smallest value is 0.</p><p><strong>c)</strong> The of <code class='latex inline'>f</code> has local extrema at <code class='latex inline'>(-1, 6)</code> and <code class='latex inline'>(3, 1)</code>.</p>
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>f(x)= 3x^3 +7x^2 + 3x -1</code></p>
<p>(a) Sketch the graph of a continuous function <code class='latex inline'>\displaystyle f </code> with all of the following properties:</p><p>(i) <code class='latex inline'>\displaystyle f(0)=2 </code></p><p>(ii) <code class='latex inline'>\displaystyle f(x) </code> is decreasing for <code class='latex inline'>\displaystyle 0 \leq x \leq 3 </code> (iii) <code class='latex inline'>\displaystyle f(x) </code> is increasing for <code class='latex inline'>\displaystyle 3 < x \leq 5 </code> (iv) <code class='latex inline'>\displaystyle f(x) </code> is decreasing for <code class='latex inline'>\displaystyle x > 5 </code> (v) <code class='latex inline'>\displaystyle f(x) \rightarrow 9 </code> as <code class='latex inline'>\displaystyle x \rightarrow \infty </code> (b) Is it possible that the graph of <code class='latex inline'>\displaystyle f </code> is concave down for all <code class='latex inline'>\displaystyle x > 6 </code> ? Explain.</p>
<p>Use the critical points to sketch each function.</p><p><code class='latex inline'>g(x) = 27x - x^3</code></p>
<p>Sketch each function using suitable techniques.</p><p><code class='latex inline'>y=x(x-4)^3</code> </p>
<p>For each function,</p><p>i) How does the function behave as <code class='latex inline'>x\to \infty</code>? Explain your reasoning.</p><p>ii) Does the function have any symmetry? Explain.</p><p>iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.</p><p>iv) Find the point(s) of inflection and make a table showing the intervals of concavity.</p><p><code class='latex inline'> \displaystyle g(x) = x^3 -27x </code></p>
<ul> <li>What is the maximum number of local extrema this function can have? Explain.</li> <li>What is the maximum number of points of inflection this function can have? Explain.</li> <li>Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.</li> <li>Sketch the function.</li> </ul> <p><code class='latex inline'>\displaystyle k(x)=-2 x^{4}+16 x^{2}-12 </code></p>
<p>Sketch a function has the following characteristics:</p> <ul> <li><code class='latex inline'>f(0) = -1.5</code></li> <li><code class='latex inline'>f(1) = 2</code></li> <li>There is a vertical asymptote at <code class='latex inline'>x = -1</code>.</li> <li>As <code class='latex inline'>x</code> gets positively large, <code class='latex inline'>y</code> gets positively large.</li> <li>As <code class='latex inline'>x</code> gets negatively large, <code class='latex inline'>y</code> approaches zero.</li> </ul>
<p>Use the critical points to sketch each function.</p><p><code class='latex inline'>g(x) = x(x + 2)^2</code></p>
<p>Find the x- and y-intercepts of each function, and then sketch the curve.</p><p><code class='latex inline'>y = x^3 + 3x^2 +1</code></p>
<p>Sketch the graph of a function <code class='latex inline'>y = f(x)</code> that satisfies each set of conditions.</p><p><code class='latex inline'>\displaystyle \lim_{x\to 5^+}f(x) = -3, \lim_{x\to 5^-}f(x) = 1</code>, and <code class='latex inline'>f(5) = 0</code></p>
<p>Sketch each function using suitable techniques.</p><p><code class='latex inline'>g(x)=\displaystyle{\frac{x^2+1}{4x^2-9}}</code></p>
<p>The set of characteristics describes a parent function that has been shifted. Draw a possible graph, and state whether the graph is continuous.</p><p>The interval of increase is <code class='latex inline'>(-\infty, \infty)</code>, and there is a horizontal asymptote at <code class='latex inline'>y =-10</code>.</p>
<p>Determine a parent function that matches each set of characteristics.</p><p>a) The graph is neither even nor odd, and as <code class='latex inline'>x \to \infty</code>, <code class='latex inline'>y \to \infty</code></p><p>b) <code class='latex inline'>(-\infty, 0)</code>, and <code class='latex inline'>(0, \infty)</code> are both intervals of decrease.</p><p>c) The domain is <code class='latex inline'>[0, \infty)</code></p>
<p>Sketch the graph of a function <code class='latex inline'>y = f(x)</code> that satisfies each set of conditions.</p><p><code class='latex inline'>\displaystyle \lim_{x\to -1^+}f(x) = 3, \lim_{x\to -1^-}f(x) = 2</code>, and <code class='latex inline'>f(-1) = 5</code></p>
<p>Is it always true that an increasing function is concave up in shape? Explain. </p>
<ul> <li>i. Find the local maximum and minimum values.</li> <li>ii. Find the intervals of concavity and the inflection points.</li> <li>iii. Summarize all information on a table as shown in the lecture.</li> <li><p>iv. Sketch the graph.</p></li> <li><p> <code class='latex inline'> f(x) = 20x^3 - 3x^5</code></p></li> </ul>
<p>Determine the critical points for each of the following functions, and determine whether the function has a local maximum value, a local minimum value, or neither at the critical points. Sketch the graph of each function.</p><p><code class='latex inline'>\displaystyle f(x)=3 x^{4}-4 x^{3} </code></p>
<p>Use the critical points to sketch each function.</p><p><code class='latex inline'>g(x) = x^4 -8x^2 -10</code></p>
<p>Graph <code class='latex inline'>f(x)</code> if <code class='latex inline'>f'(x) < 0</code> when <code class='latex inline'>x < -2</code> and <code class='latex inline'>x >3, f'(x) > 0</code> when <code class='latex inline'>-2< x < 3, f(-2) =0</code>, and <code class='latex inline'>f(3) = 5</code>.</p>
<p>Consider the function <code class='latex inline'>\displaystyle g(x)=\frac{1}{x^{2}-1} </code></p><p>a) How does <code class='latex inline'>\displaystyle g(x) </code> behave as <code class='latex inline'>\displaystyle x \rightarrow \pm \infty </code> ? Explain your reasoning.</p><p>b) State equations for the vertical asymptotes.</p><p>c) Evaluate the one-sided limits at one of the asymptotes.</p><p>d) What does symmetry tell you about the other asymptote?</p><p>e) Find and classify the critical point(s).</p>
<p>Sketch graphs of functions with the following characteristics.</p><p>A nonlinear graph has <code class='latex inline'>\displaystyle x </code> -intercepts at <code class='latex inline'>\displaystyle -8 </code> and <code class='latex inline'>\displaystyle -2 </code> and a <code class='latex inline'>\displaystyle y </code> -intercept at 3 . The graph has relative minimums at <code class='latex inline'>\displaystyle x=-6 </code> and <code class='latex inline'>\displaystyle x=6 </code> and a relative maximum at <code class='latex inline'>\displaystyle x=2 </code>. The graph is positive for <code class='latex inline'>\displaystyle x < -8 </code> and <code class='latex inline'>\displaystyle x > -2 </code> and negative between <code class='latex inline'>\displaystyle x=-8 </code> and <code class='latex inline'>\displaystyle x=-2 </code>. As <code class='latex inline'>\displaystyle x </code> decreases, <code class='latex inline'>\displaystyle y </code> increases and as <code class='latex inline'>\displaystyle x </code> increases, <code class='latex inline'>\displaystyle y </code> increases.</p>
<p>Sketch graphs of functions with the following characteristics.</p><p>A nonlinear graph has <code class='latex inline'>\displaystyle x </code> -intercepts at <code class='latex inline'>\displaystyle -2 </code> and 2 and a <code class='latex inline'>\displaystyle y </code> -intercept at <code class='latex inline'>\displaystyle -4 </code>. The graph has a relative minimum of <code class='latex inline'>\displaystyle -4 </code> at <code class='latex inline'>\displaystyle x=0 </code>. The graph is decreasing for <code class='latex inline'>\displaystyle x < 0 </code> and increasing for <code class='latex inline'>\displaystyle x > 0 </code>.</p>
<p>A function has vertical asymptotes at <code class='latex inline'>\displaystyle x=-1 </code> and <code class='latex inline'>\displaystyle x=3 </code> and approaches zero as <code class='latex inline'>\displaystyle x </code> approaches <code class='latex inline'>\displaystyle \pm \infty </code>. It is concave down on the intervals <code class='latex inline'>\displaystyle -\infty < x < -1 </code> and <code class='latex inline'>\displaystyle 1 < x < 3 . </code> It is concave up on the intervals <code class='latex inline'>\displaystyle -1 < x < 1 </code> and <code class='latex inline'>\displaystyle 3 < x < \infty </code></p><p>a) Sketch a possible graph of such a function.</p><p>b) Is there more than one equation that could model such a graph? Justify your answer.</p><p>c) Is it possible to have any local extrema between <code class='latex inline'>\displaystyle -1 </code> and 3 ? Justify your answer.</p>
<p>For the function <code class='latex inline'>\displaystyle g(x)=\frac{1}{x^{2}+1} </code>, answer all of the questions without finding derivatives.</p><p>a) Find all of the intercepts.</p><p>b) Find the maximum value. Explain your reasoning.</p><p>c) State the equation of the horizontal asymptote.</p><p>d) Are there any local minima? Use your answers to parts a) through c) to explain. e) How many points of inflection are there? Explain your reasoning.</p><p>f) Sketch the graph, and then verify your work using a graphing calculator.</p>
<p>For the function <code class='latex inline'>f(x) = 2x^3 - x^4</code>, determine the critical points and classify them using the second derivative test. Sketch the function.</p>
<p>Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.</p><p>The polynomial function g has domain <code class='latex inline'>\mathbb{R}</code>.</p><p><code class='latex inline'>\displaystyle \lim_{x\to -\infty} g(x) = \infty </code></p><p><code class='latex inline'>g</code> is an odd function.</p><p>There are five x-intercepts, two of which are 4 and -22. The function passes through <code class='latex inline'>( 1, 5)</code>, which is close to a local extremum.</p>
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>f(x)= 2x^4 -26x^2 +72</code></p>
<p>If <code class='latex inline'>f(x)</code> is an absolute value function that has the characteristics below?</p> <ul> <li>The graph of <code class='latex inline'>f(x)</code> decreases on the interval <code class='latex inline'>(-\infty, -2)</code> and </li> <li>increases on the interval (2, <code class='latex inline'>\infty</code>).<br></li> <li>It has a <code class='latex inline'>y-</code>intercept (0, 4). What is possible equation for <code class='latex inline'>f(x)</code>?<br></li> </ul> <p>Is there only one such function?</p>
<p>For <code class='latex inline'>f(x)=\displaystyle{\frac{5x}{(x-1)^2}}</code>, show that <code class='latex inline'>f'(x)=\displaystyle{\frac{-5(x+1)}{(x-1)^3}}</code> and <code class='latex inline'>f''(x)=\displaystyle{\frac{100(x+2)}{(x-1)^4}}</code>. Use the function and its derivatives to determine the domain, intercepts, asymptotes, intervals of increase and decrease, and concavity, and to locate any local extrema and points of inflection. Use this information to sketch the graph of <code class='latex inline'>f</code>.</p>
<p>Find the equation of the tangent to the curve <code class='latex inline'>\displaystyle y=5^{-x^{2}} </code> at the point on the curve where <code class='latex inline'>\displaystyle x=1 </code>. Graph the curve and the tangent at this point.</p>
<p>The function <code class='latex inline'>\displaystyle f(x)=-x+\frac{1}{x^{2}} </code> can be considered a sum of two functions.</p><p>a) Identify the two functions.</p><p>b) Sketch the two functions lightly on the same set of axes.</p><p>c) Use your results from part b) to predict what the graph of <code class='latex inline'>\displaystyle f(x) </code> will look like.</p><p>d) Verify your prediction using graphing technology.</p><p>e) Find the first derivative of <code class='latex inline'>\displaystyle f(x)=-x+\frac{1}{x^{2}} </code> and use it to find the turning point.</p><p>f) Find the second derivative of <code class='latex inline'>\displaystyle f(x)=-x+\frac{1}{x^{2}} </code> and use it to find the intervals of concavity.</p>
<p>Sketch each function.</p><p><code class='latex inline'>b(x)=-(2x-1)(x^2-x-2)</code></p>
<p>Analyse and sketch the function.</p><p><code class='latex inline'>\displaystyle f(x) = \frac{1}{4}x^4 - \frac{9}{2}x^2 </code></p>
<p>For each function,</p><p>i) How does the function behave as <code class='latex inline'>x\to \infty</code>? Explain your reasoning.</p><p>ii) Does the function have any symmetry? Explain.</p><p>iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.</p><p>iv) Find the point(s) of inflection and make a table showing the intervals of concavity.</p><p><code class='latex inline'> \displaystyle y = x^4 - 8x^2 + 16 </code></p>
<p>Consider the function <code class='latex inline'>\displaystyle f(x)=\frac{x-1}{x+1} </code></p><p>a) Find the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-intercepts of its graph.</p><p>b) State equations for the asymptotes.</p><p>c) Find the first derivative and use it to determine when the function is increasing and decreasing.</p><p>d) Explain how your answer to part c) helps determine the behaviour of the function near the asymptotes.</p><p>e) Sketch the graph.</p>
<p>Sketch each function using suitable techniques.</p><p><code class='latex inline'>h(x)=\displaystyle{\frac{x}{x^2-4x+4}}</code></p>
<p>Analyse and sketch the function.</p><p><code class='latex inline'>\displaystyle f(x) = -x^2 +2x </code></p>
<p>Consider the function <code class='latex inline'>f(x)=2x^3-3x^2-72x+7</code></p> <ul> <li>What is the maximum number of local extrema this function can have? Explain.</li> <li>What is the maximum number of points of inflection this function can have? Explain.</li> <li>Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.</li> <li>Sketch the function.</li> </ul>
<ul> <li>What is the maximum number of local extrema this function can have? Explain.</li> <li>What is the maximum number of points of inflection this function can have? Explain.</li> <li>Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.</li> <li>Sketch the function.</li> </ul> <p><code class='latex inline'>\displaystyle g(x)=x^{4}-8 x^{2}+16 </code></p>
<p>Analyse and sketch the function.</p><p><code class='latex inline'>\displaystyle f(x) = 2x^3 -3x^2 -3x + 2 </code></p>
<p>Sketch the graph of a function <code class='latex inline'>y = f(x)</code> that satisfies each set of conditions.</p><p><code class='latex inline'>\displaystyle \lim_{x\to 2^+}f(x) = -1, \lim_{x\to 2^-}f(x) = -4</code>, and <code class='latex inline'>f(2) = -4</code></p>
<p>Sketch each function using suitable techniques. </p><p><code class='latex inline'>y=x^4-8x^2+7</code></p>
<p>Sketch ha graph of a function f with all of the following properties:</p> <ul> <li>The graph is increasing when <code class='latex inline'>x < -2</code> and when $- 2&lt; x &lt; 4$`.</li> <li>The grap his decreasing when <code class='latex inline'>x > 4</code>.</li> <li><code class='latex inline'>f'(-2) =0, f'(4) = 0</code></li> <li>The graph his concave down when <code class='latex inline'>x < -2</code> and when <code class='latex inline'>3 < x < 9</code>.</li> <li>The graph his concave up when <code class='latex inline'>-2 < x < 3</code> and when <code class='latex inline'>x > 9</code>.</li> </ul>
<ul> <li><code class='latex inline'>f(x)</code> is quadratic function. The graph of <code class='latex inline'>f(x)</code> decreases on the interval <code class='latex inline'>(-\infty, -2)</code> and increases on the interval (2, <code class='latex inline'>\infty</code>). It has a <code class='latex inline'>y-</code>intercept (0, 4). What is possible equation for <code class='latex inline'>f(x)</code>?<br></li> <li>Is there only one quadratic function, <code class='latex inline'>f(x)</code>, that has the characteristics given in part (a)?</li> </ul>
<p>Sketch graphs of functions with the following characteristics.</p><p>The graph is linear with an <code class='latex inline'>\displaystyle x </code> -intercept at <code class='latex inline'>\displaystyle -2 </code>. The graph is positive for <code class='latex inline'>\displaystyle x < -2 </code>, and negative for <code class='latex inline'>\displaystyle x > -2 </code>.</p>
<p>Sketch each function.</p><p><code class='latex inline'>f(x)=x^3+1</code></p>
<p>Use the critical points to sketch each function.</p><p><code class='latex inline'>f(x) = 7 + 6x -x^2</code></p>
<p>Use at least five curve-sketching techniques to explain how to sketch the graph of the function <code class='latex inline'>\displaystyle f(x) = \frac{2x + 10}{x^2 -9} </code>. Sketch the graph on graph paper.</p>
<p>Do a full sketch by showing how you got</p> <ul> <li>Horizontal and vertical Asymptotes</li> <li>x-intercepts</li> <li>Any critical points (max/min)</li> <li>Intervals of increase and decrease</li> <li>Points of inflection and concavity</li> </ul> <p><code class='latex inline'>f(x)= 2x^3 -12x^2 + 18x -4</code></p>
<p>For each function,</p><p>i) How does the function behave as <code class='latex inline'>x\to \infty</code>? Explain your reasoning.</p><p>ii) Does the function have any symmetry? Explain.</p><p>iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.</p><p>iv) Find the point(s) of inflection and make a table showing the intervals of concavity.</p><p><code class='latex inline'> \displaystyle k(x) = \frac{1}{1 -x^2} </code></p>
<p>For each function,</p><p>i) How does the function behave as <code class='latex inline'>x\to \infty</code>? Explain your reasoning.</p><p>ii) Does the function have any symmetry? Explain.</p><p>iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.</p><p>iv) Find the point(s) of inflection and make a table showing the intervals of concavity.</p><p><code class='latex inline'>h(x) = \frac{x-4}{x^2}</code></p>
<p>Sketch a graph that has the following characteristics:</p> <ul> <li>The function is odd.</li> <li>The function is continuous.</li> <li>The function has zeros at <code class='latex inline'>x = -3 0</code>, and <code class='latex inline'>3</code>.</li> <li>The function is increasing on the intervals <code class='latex inline'>x\in (-\infty, -2)</code> or <code class='latex inline'>x\in (2, \infty)</code></li> <li>The function is decreasing on the interval <code class='latex inline'>x\in(-2, 2)</code>.</li> </ul>
<p>Sketch the graph of a function with the following properties:</p> <ul> <li>There are local extrema at (-1, 7) and (3, 2).</li> <li>There is a point of inflection at (1, 4).</li> <li>The graph is concave down only when <code class='latex inline'>x <1</code>.</li> <li>The x-intercept is -4 and the y-intercept is 6.</li> </ul>
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