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Mathematics LibreTexts

Test 2

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  • Page ID
    10986
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    These mock exams are provided to help you prepare for  Term/Final tests.  The best way to use these practice tests is to try the problems as if you were taking the test. Please don't look at the solution until you have attempted the question(s). Only reading through the answers or studying them, will typically not be helpful in preparing since it is too easy to convince yourself that you understand it.

    Mock Exam (Test 2) 

    You can try timing yourself for 50 minutes.

     

    Exercise \(\PageIndex{1}\): definition of the derivative

    a) Use the definition of the derivative to find \(f'(x)\) for

    $$\displaystyle{f(x)=\frac{1}{3x+1}}$$

    Hint:

    The definition of the derivative \(f'(x)= \lim_{h \rightarrow 0} \displaystyle\frac{f(x+h)-f(x)}{h}\).

    Answer:

    \(\displaystyle{f(x)=\frac{-3}{(3x+1)^2}}\).

    Solution:

    \(f'(x)= \lim_{h \rightarrow 0} \displaystyle\frac{f(x+h)-f(x)}{h}\).

     

    b) Find the slope of the tangent line to the graph at the point \(x=1.\)

    Answer

    \(\displaystyle{f(x)=\frac{-3}{16}}\).

    Exercise \(\PageIndex{2}\): L'Hopital's

    Find the following limits:

    1. \(\displaystyle \lim_{x→0^+}x^2lnx\)
    2. \( \displaystyle \lim_{x→π}\frac{1+cosx}{sinx}\)
    3. \(\displaystyle \lim_{x\rightarrow 0 }\left(e^x + x\right)^{1/x}.\)

    Answer:

    \(0. 0, e^2\)

     

     

    Exercise \(\PageIndex{3}\): derivatives

    Find the derivative  \(\displaystyle{\frac{dy}{dx}}\) of the following functions/relations:

    1. \(y(x)=\displaystyle \frac{ln(x^2+1)}{x^2}\). Note that \(y(x)\) is \(y\) as a function of \(x\).

    2. \(\displaystyle{y=\sin^{-1}(x^3)}\).

    3. \(x^2y+xy+y^2=1\)

    4. \(y(x)= x^{\sin x}\)

    Answer:

    \( \displaystyle\frac{2}{x^2+x} -\displaystyle \frac{2 \ln(x^2+1)}{x^3}, \displaystyle\frac{3x^2}{\sqrt{1-x^6}},  \displaystyle\frac{-(2xy+y)}{x^2+x+2y},  x^{\sin x-1}(\sin(x)+x \ln(x) \cos(x))\).

     

    Solution:

    1.    \(\displaystyle \frac{dy}{dx}=\displaystyle \frac{\displaystyle \frac{2x}{x^2+1}-(2x)ln(x^2+1)x^2 }{x^4},\) by quotient rule,

    \(= \displaystyle \frac{2}{x(x^2+1)}-\displaystyle \frac{2ln(x^2+1)}{x^3},\) 

    \(= \displaystyle\frac{2}{x^2+x}- \displaystyle \frac{2 \ln(x^2+1)}{x^3}.\)

    4.  Let \(y=x^{\sin x}\).

    Then \( ln(y)= ln \left(x^{\sin x}  \right)\).

    Which implies,   \( ln(y)=  \sin(x) ln \left(x\right)\).

    Differentiate with respect to \(x\) both sides,

    \( \displaystyle \frac{1}{y} (\displaystyle \frac{dy}{dx}=  \cos(x) \ln(x)+ \sin(x) \displaystyle \frac{1}{x}\),

    \(  (\displaystyle \frac{dy}{dx}=  y\left(\cos(x) \ln(x)+ \sin(x) \displaystyle \frac{1}{x} \right)\),

    \(  (\displaystyle \frac{dy}{dx}=  x^{\sin x}\left(\cos(x) \ln(x)+ \sin(x) \displaystyle \frac{1}{x} \right)\), Hence the result.

     

     

    Exercise \(\PageIndex{4}\): Related rates

    A spherical balloon is being inflated at a rate of \(2 m^3/min.\) Find how fast the surface area of the balloon is increasing when \(r=5m.\)

    (The surface area of the sphere is \(4\pi r^2\) and the volume \(v=\displaystyle \frac{4}{3} \pi r^3,\) where \(r\) is the radius of the sphere.)

    Answer

    \(\displaystyle\frac{4}{5} m^2/min\).

     

    Exercise \(\PageIndex{5}\): 

    Consider the function \(f(x)=x^5 - 5x^4\).

    1.  Find the intervals on which \(f\) is increasing.
    2.  Find the intervals on which \(f\)  is decreasing.
    3.  Find the value of the relative minima(s)( if any) of the function.
    4. Find the value of the relative maxima(s)( if any) of the function.
    5.  Find the open intervals on which \(f\)  is concave up.
    6. Find the open intervals on which \(f\)  is concave down.
    7. Find the \(x-\) coordinates of all inflection points.
    Answer

    \((-\infty, 0)\cup (4, \infty), (0,4), x=4, x=0,(3,\infty), (-\infty,3). x=3\)

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