5.6 Chapter Review Log and Trig Functions
Chapter
Chapter 5
Section
5.6
Solutions 56 Videos

Differentiate each of the following:

y=6-e^x

0.11mins
Q1a

Differentiate each of the following:

y=2x+3e^x

0.17mins
Q1b

Differentiate each of the following:

y=e^{2x+3}

0.15mins
Q1c

Differentiate each of the following:

y=e^{-3x^2+5x}

0.26mins
Q1d

Differentiate each of the following:

y=xe^x

0.17mins
Q1e

Differentiate each of the following:

s=\displaystyle{\frac{e^t-1}{e^t+1}}

0.41mins
Q1f

Determine $\displaystyle{\frac{dy}{dx}}$ for each of the following:

y=10^x

0.14mins
Q2a

Determine \displaystyle{\frac{dy}{dx}} for each of the following:

y=4^{3x^2}

0.17mins
Q2b

Determine \displaystyle{\frac{dy}{dx}} for each of the following:

y=(5x)(5^x)

0.38mins
Q2c

Determine \displaystyle{\frac{dy}{dx}} for each of the following:

y=(x^4)2^x

0.28mins
Q2d

Determine \displaystyle{\frac{dy}{dx}} for each of the following:

y=\displaystyle{\frac{4x}{4^x}}

0.48mins
Q2e

Determine \displaystyle{\frac{dy}{dx}} for each of the following:

y=\displaystyle{\frac{5^{\sqrt{x}}}{x}}

1.00mins
Q2f

Differentiate each of the following:

y=3\sin2x-4\cos2x

0.35mins
Q3a

Differentiate each of the following:

y=\tan3x

0.12mins
Q3b

Differentiate each of the following:

y=\displaystyle{\frac{1}{2-\cos x}}

0.24mins
Q3c

Differentiate each of the following:

y=x\tan2x

0.18mins
Q3d

Differentiate each of the following:

y=(\sin2x)e^{3x}

0.21mins
Q3e

Differentiate each of the following:

y=\cos^22x

0.34mins
Q3f

a) Given the function f(x)=\displaystyle{\frac{e^x}{x}}, solve the equation f'(x)=0.

b) Discuss the significance of the solution you found in part a.

0.25mins
Q4

If f(x)=xe^{-2x}, find f'\left(\displaystyle{\frac{1}{2}}\right).

0.44mins
Q5a

a) If f(x)=xe^{-2x}, find f'\left(\displaystyle{\frac{1}{2}}\right).

b) Explain what this number represents.

0.11mins
Q5b

Determine the second derivative of each of the following:

y=xe^x-e^x

0.26mins
Q6a

Determine the second derivative of

y=xe^{10x}

0.21mins
Q6b

If y=\displaystyle{\frac{e^{2x}-1}{e^{2x}+1}}, \displaystyle{\frac{dy}{dx}}= ?

2.37mins
Q7

Determine the equation of the tangent to the curve defined by y=x-e^{-x} that is parallel to the line represented by 3x-y-9=0.

1.23mins
Q8

Determine the equation of the tangent to the curve y=x\sin x at the point where x=\displaystyle{\frac{\pi}{2}}.

0.44mins
Q9

An object moves along a line so that, at time t, its position is s=\dfrac{\sin t}{3+\cos2t}, where s is the displacement in metres. Calculate the object's velocity at t=\displaystyle{\frac{\pi}{4}}.

2.02mins
Q10

The number of bacteria in a culture, N, at time t is given by N(t)=2000[30+te^{-\displaystyle{\frac{t}{20}}}].

• When is the rate of change of the number of bacteria equal to zero?
1.01mins
Q11a

The number of bacteria in a culture, N, at time t is given by N(t)=2000[30+te^{-\displaystyle{\frac{t}{20}}}].

• If the bacterial culture is placed in a colony of mice, the number of mice that become infected, M, is related to the number of bacteria present by the equation M(t)=\sqrt{N+1000}. After 10 days, how many mice are infected per day?
1.13mins
Q11b

The concentrations of two medicines in the bloodstream t hours after injections are c_{1}(t)=te^{-1} and c_{2}(t)=t^2e^{-t}.

• Which medicine has the larger maximum concentration?
1.21mins
Q12a

The concentrations of two medicines in the bloodstream t hours after injections are c_{1}(t)=te^{-1} and c_{2}(t)=t^2e^{-t}.

• Within the first half hour, which medicine has the larger maximum concentration?
0.56mins
Q12b

Differentiate.

y=(2+3e^{-x})^3

0.32mins
Q13a

Differentiate.

y=x^e

0.15mins
Q13b

Differentiate.

y=e^{e^x}

0.14mins
Q13c

Differentiate.

y=(1-e^{5x})^5

0.26mins
Q13d

Differentiate.

y=5^x

0.08mins
Q14a

Differentiate.

y=(0.47)^x

0.11mins
Q14b

Differentiate.

y=(52)^{2x}

0.15mins
Q14c

Differentiate.

y=5(2)^x

0.17mins
Q14d

Differentiate.

y=4(e)^x

0.13mins
Q14e

Differentiate.

y=-2(10)^{3x}

0.15mins
Q14f

Determine y'.

y=\sin2^x

0.14mins
Q15a

Determine y'.

y=x^2\sin x

0.13mins
Q15b

Determine $y'$.

y=\sin \left( \displaystyle{\frac{\pi}{2}}-x \right)

0.14mins
Q15c

Determine y'.

y=\cos x\sin x

0.27mins
Q15d

Determine $y'$.

y=\cos^2x

0.23mins
Q15e

Determine y'.

y=\cos x

0.44mins
Q15f

Determine the equation of the tangent to the curve y=\cos x at \left(\displaystyle{\frac{\pi}{2}},0 \right) .

0.27mins
Q16

An object is suspended from the end of a spring. Its displacement from the equilibrium position is s=8\sin(10\pi t) at time t. Calculate the velocity and acceleration of the object at any time t, and show that \displaystyle{\frac{d^2s}{dt^2}}+100\pi^2s=0.

1.22mins
Q17

The position of a particle is given by s=5\cos\left(2t+\displaystyle{\frac{\pi}{4}}\right) at time t. What are the maximum values of the displacement, the velocity, and the acceleration?

1.57mins
Q18

The hypotenuse of a right triangle is 12 cm in length. Calculate the measure of the unknown angles in the triangle that will maximize its perimeter. 2.09mins
Q19

A fence is 1.5 m high and is 1 m from a wall. A ladder must start from the ground, touch the top of the fence, and rest somewhere on the wall. Calculate the minimum length of the ladder.

5.28mins
Q20

A thin rigid pole needs to be carried horizontally around a corner joining two corridors, which are 1 m and 0.8 m wide. Calculate the length of the longest pole that can be carried around this corner. 4.20mins
Q21

When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feed (slightly less than 2 m). If Sidney Crosby is on the goal line, three feet outside on of the goal posts, how far should he go out (perpendicular to the goal line) to maximize the angle in which he can shoot at the goal? Hint: Determine the values of x that maximize \theta in the following diagram. 6.17mins
Q22

Determine f''(x)

f(x)=4\sin^2(x-2)

Determine f''(x)
f(x)=2(\cos x)(\sec^2x)