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Subsection 2.9.4 Exercises

Exercises — Stage 1


Suppose the amount of kelp in a harbour depends on the number of urchins. Urchins eat kelp: when there are more urchins, there is less kelp, and when there are fewer urchins, there is more kelp. Suppose further that the number of urchins in the harbour depends on the number of otters, who find urchins extremely tasty: the more otters there are, the fewer urchins there are.

Let \(O\text{,}\) \(U\text{,}\) and \(K\) be the populations of otters, urchins, and kelp, respectively.

  1. Is \(\diff{K}{U}\) positive or negative?
  2. Is \(\diff{U}{O}\) positive or negative?
  3. Is \(\diff{K}{O}\) positive or negative?

Remark: An urchin barren is an area where unchecked sea urchin grazing has decimated the kelp population, which in turn causes the other species that shelter in the kelp forests to leave. Introducing otters to urchin barrens is one intervention to increase biodiversity. A short video with a more complex view of otters and urchins in Canadian waters is available on YouTube:


Suppose \(A, B, C, D\) and \(E\) are functions describing an interrelated system, with the following signs: \(\diff{A}{B} \gt 0\text{,}\) \(\diff{B}{C} \gt 0\text{,}\) \(\diff{C}{D} \lt 0\text{,}\) and \(\diff{D}{E} \gt 0\text{.}\) Is \(\diff{A}{E}\) positive or negative?

Exercises — Stage 2


Evaluate the derivative of \(f(x)=\cos(5x+3)\text{.}\)


Evaluate the derivative of \(f(x)=\left({x^2+2}\right)^5\text{.}\)


Evaluate the derivative of \(T(k)=\left({4k^4+2k^2+1}\right)^{17}\text{.}\)


Evaluate the derivative of \(f(x)=\sqrt{\dfrac{x^2+1}{x^2-1}}\text{.}\)


Evaluate the derivative of \(f(x)=e^{\cos(x^2)}\text{.}\)

8 (✳)

Evaluate \(f'(2)\) if \(f(x) = g\big(x/h(x)\big)\text{,}\) \(h(2) = 2\text{,}\) \(h'(2) = 3\text{,}\) \(g'(1) = 4\text{.}\)

9 (✳)

Find the derivative of \(e^{x\cos(x)}\text{.}\)

10 (✳)

Evaluate \(f'(x)\) if \(f(x) = e^{x^2+\cos x}\text{.}\)

11 (✳)

Evaluate \(f'(x)\) if \(f(x) = \sqrt{\dfrac{x-1}{x+2}}\text{.}\)

12 (✳)

Differentiate the function

\begin{equation*} f(x)=\frac{1}{x^2}+\sqrt{x^2-1} \end{equation*}

and give the domain where the derivative exists.

13 (✳)

Evaluate the derivative of \(f(x)=\dfrac{\sin 5x}{1+x^2}\)


Evaluate the derivative of \(f(x)=\sec(e^{2x+7})\text{.}\)


Find the tangent line to the curve \(y=\left(\tan^2 x +1\right)\left(\cos^2 x\right)\) at the point \(x=\dfrac{\pi}{4}\text{.}\)


The position of a particle at time \(t\) is given by \(s(t)=e^{t^3-7t^2+8t}\text{.}\) For which values of \(t\) is the velocity of the particle zero?


What is the slope of the tangent line to the curve \(y=\tan\left(e^{x^2}\right)\) at the point \(x=1\text{?}\)

18 (✳)

Differentiate \(y=e^{4x}\tan x\text{.}\) You do not need to simplify your answer.

19 (✳)

Evaluate the derivative of the following function at \(x=1\text{:}\) \(f(x)=\dfrac{x^3}{1+e^{3x}}\text{.}\)

20 (✳)

Differentiate \(e^{\sin^2(x)}\text{.}\)

21 (✳)

Compute the derivative of \(y=\sin\left(e^{5x}\right)\)

22 (✳)

Find the derivative of \(e^{\cos(x^2)}\text{.}\)

23 (✳)

Compute the derivative of \(y=\cos\big(x^2+\sqrt{x^2+1}\big)\)

24 (✳)

Evaluate the derivative.

\begin{equation*} y=(1+x^2)\cos^2 x \end{equation*}
25 (✳)

Evaluate the derivative.

\begin{equation*} y=\frac{e^{3x}}{1+x^2} \end{equation*}
26 (✳)

Find \(g'(2)\) if \(g(x)=x^3h(x^2)\text{,}\) where \(h(4)=2\) and \(h'(4)=-2\text{.}\)

27 (✳)

At what points \((x,y)\) does the curve \(y=xe^{-(x^2-1)/2}\) have a horizontal tangent?


A particle starts moving at time \(t=1\text{,}\) and its position thereafter is given by

\begin{equation*} s(t)=\sin\left(\frac{1}{t}\right). \end{equation*}

When is the particle moving in the negative direction?


Compute the derivative of \(f(x)=\dfrac{e^{x}}{\cos^3 (5x-7)}\text{.}\)

30 (✳)

Evaluate \(\ds\diff{}{x}\left\{x e^{2x} \cos 4x\right\}\text{.}\)

Exercises — Stage 3


A particle moves along the Cartesian plane from time \(t=-\pi/2\) to time \(t=\pi/2\text{.}\) The \(x\)-coordinate of the particle at time \(t\) is given by \(x=\cos t\text{,}\) and the \(y\)-coordinate is given by \(y=\sin t\text{,}\) so the particle traces a curve in the plane. When does the tangent line to that curve have slope \(-1\text{?}\)

32 (✳)

Show that, for all \(x \gt 0\text{,}\) \(e^{x+x^2} \gt 1+x\text{.}\)


We know that \(\sin (2x) = 2\sin x \cos x\text{.}\) What other trig identity can you derive from this, using differentiation?


Evaluate the derivative of \(f(x)=\sqrt[3]{\dfrac{e^{\csc x^2}}{ \sqrt{x^3-9} \tan x }}\text{.}\) You do not have to simplify your answer.


Suppose a particle is moving in the Cartesian plane over time. For any real number \(t \geq 0\text{,}\) the coordinate of the particle at time \(t\) is given by \((\sin t, \cos^2 t)\text{.}\)

  1. Sketch a graph of the curve traced by the particle in the plane by plotting points, and describe how the particle moves along it over time.
  2. What is the slope of the curve traced by the particle at time \(t=\dfrac{10\pi}{3}\text{?}\)