Textbook

Calculus and Vectors Nelson
Chapter

Chapter 5
Section

Chapter Test on Logs and Trig Derivatives

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Solutions
15 Videos

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = e^{-2x^2}
```

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Q1a

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = 3^{x^2 + 3x}
```

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Q1b

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = \frac{3^{3x} + e^{-3x}}{2}
```

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Q1c

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = 2\sin x -3\cos 5x
```

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Q1d

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = \sin^3(x^2)
```

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Q1e

Determine `\frac{dy}{dx}`

.

```
\displaystyle
y = \tan \sqrt{1-x}
```

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Q1f

Determine the equation of the tangent to the curve defined by `y = 2e^{3x}`

that is parallel to the line defined by `-6x + y =2`

.

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Q2

Determine the equation of the tangent to `y = e^x + \sin x`

at `(0, 1)`

.

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Q3

The velocity of a certain particle that moves in a straight line under the influence of forces is given by `v(t) = 10e^{-kt}`

, where `k`

is a positive constant and `v(t)`

is in centimetres per second.

a) Show that the acceleration of the particle is proportional to a constant multiple of its velocity. Explain what is happening to the particle.

b) What is the initial velocity of the particle?

c) At what time is the velocity equal to half the initial velocity? What is the acceleration at this time?

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Q4

Determine `f''(x)`

.

```
\displaystyle
f(x) = \cos^2x
```

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Q5a

Determine `f''(x)`

.

```
\displaystyle
f(x) = \cos x \cot x
```

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Q5b

Determine the absolute extreme values of `f(x) = \sin^2x`

, where `x \in [0, \pi]`

.

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Q6

Calculate the slope of the tangent line that passes through `y = 5^x`

, where `x = 2`

. Express your answer to two decimal places.

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Q7

Determine all the maximum and minimum values of `y = xe^x + 3ex`

.

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Q8

`f(x) = 2\cos x - \sin 2x`

where `x \in [-\pi, \pi]`

.

a) Determine all critical number for `f(x)`

on the given interval.

b) Determine the intervals where `f(x)`

is increasing and where it is decreasing.

c) Determine all local maximum and minimum value of `f(x)`

on the given interval.

d) Use the information you found above to sketch the curve.

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Q9