Practice: derivatives and investigation

Exponential and logarithmic functions, grade 12 (4 units) Lesson 16 of 20Practice

Practice: derivatives and investigation

Differentiate.

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The function $\displaystyle {f(x)}={e^x-2x}$ is given.

Calculate the slope of the tangent to the graph at the point where $\displaystyle {x}={0}$ .

$\displaystyle f(x)=e^{x}-2x$

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$\displaystyle {f(0)}={1}$ .
Write the equation of the tangent to the graph at the point $\displaystyle (0,1)$ .

\displaystyle f(x)=e^{x}-2x

\displaystyle (0,1)

The function $\displaystyle {f(x)}={(x-2)e^x}$ is given.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=(x-2)e^{x}$

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Find the extremum point of the function, and state its kind.

$\displaystyle f(x)=(x-2)e^{x}$

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Find where the function increases.

Mark on the graph the extremum point and the points where it meets the axes.

\displaystyle f(x)=(x-2)e^{x}

The function $\displaystyle {f(x)}={\frac{\ln x}{x}}$ is given.

What is the domain of the function?

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{\ln x}{x}$

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Find the extremum point of the function, and state its kind.

Mark on the graph the extremum point and the point where it meets the $\displaystyle x$ -axis.

\displaystyle f(x)=\frac{\ln x}{x}

The function $\displaystyle {f(x)}={\ln(x^2-4x+5)}$ is given.

Find the domain of the function: solve an inequality.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\ln(x^{2}-4x+5)$

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Mark on the graph the extremum point of the function.

\displaystyle f(x)=\ln(x^{2}-4x+5)

The function $\displaystyle {f(x)}={e^{-x^2}}$ is given.

Prove that the function is even. What does that mean for its graph?

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=e^{-x^{2}}$

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Mark on the graph the extremum point of the function.
What is the horizontal asymptote of the graph?

\displaystyle f(x)=e^{-x^{2}}

The function $\displaystyle {f(x)}={\ln(x^2+a)}$ is given, $\displaystyle a$ a positive parameter.
The slope of the tangent to the graph of the function at the point where $\displaystyle {x}={1}$ is 1.

Calculate $\displaystyle a$ .

Put in the $\displaystyle a$ you found.
Write the equation of the tangent to the graph at the point where $\displaystyle {x}={1}$ .

\displaystyle f(x)=\ln(x^{2}+1)

The function $\displaystyle {f(x)}={x-\ln x}$ is given.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x-\ln x$

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Find the extremum point of the function, and state its kind.

Mark the extremum point on the graph.
What is the vertical asymptote of the graph?

\displaystyle f(x)=x-\ln x

The function $\displaystyle {f(x)}={2\ln(x^2-9)}$ is given.

Find the domain of the function.
What are the vertical asymptotes of the graph?

Find $\displaystyle f'(x)$ .
Where does the function increase, and where does it decrease?

$\displaystyle f(x)=2\ln(x^{2}-9)$

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Find the points where the graph of the function meets the $\displaystyle x$ -axis.

Mark these points on the graph.
Is the function even, odd, or neither? Explain.

\displaystyle f(x)=2\ln(x^{2}-9)

The drawing shows the graphs of two functions, $\displaystyle p$ and $\displaystyle q$ .
One of them is $\displaystyle {f(x)}={(\ln x)^2-4}$ , and the other is the derivative $\displaystyle f'(x)$ .

\displaystyle p(x)=\,?

\displaystyle q(x)=\,?

Which is $\displaystyle f$ , and which is $\displaystyle f'$ ? Explain.

Find the points where the graph of the function $\displaystyle f$ meets the $\displaystyle x$ -axis.

Formulas: exponential and logarithmic functions, grade 12 (4 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$