Review: exponential and logarithmic functions

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

Review: exponential and logarithmic functions

Solve.

Solve. Mind the domain.

A new car is worth 120,000 ₪.
Each year its value falls by 12% of its value the year before.

What will the car be worth after 5 years? Round to the agora.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

After how many years will the car first be worth less than half its value at the start? Solve with a logarithm, and round to 2 decimal places.

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The car’s value also falls at a constant rate from month to month.
Calculate the monthly decay factor, rounded to 4 decimal places.
By what percent does the car’s value fall each month?

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The function $\displaystyle {f(x)}={(x+a)e^x}$ is given; $\displaystyle a$ is a parameter.

Find $\displaystyle f'(x)$ . Take out a common factor.

$\displaystyle f(x)=(x+a)e^{x}$

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The tangent to the graph at the point where $\displaystyle {x}={0}$ is parallel to the line $\displaystyle {y}={3x}$ .
Find $\displaystyle a$ .

Put $\displaystyle {a}={2}$ and answer the remaining parts.
Find the extremum point of the function and its kind, and where the function increases and decreases.

$\displaystyle f(x)$ has the horizontal asymptote $\displaystyle {y}={0}$ .
Mark on the graph the points where it meets the axes, and the extremum point.

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

\displaystyle y=0

Show that $\displaystyle {F(x)}={(x+1)e^x}$ is an antiderivative of $\displaystyle f(x)$ : differentiate $\displaystyle F(x)$ .

$\displaystyle F(x)=(x+1)e^{x}$

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Calculate the area bounded by the graph of $\displaystyle f(x)$ , the $\displaystyle x$ -axis and the $\displaystyle y$ -axis.

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

What is the domain of the function? Write the equation of its vertical asymptote.

Find $\displaystyle f'(x)$ .

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

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Find the extremum point of the function and its kind, and where the function increases and decreases.

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

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

\displaystyle x=0

Show that $\displaystyle {F(x)}={\frac{(\ln x)^2}{2}}$ is an antiderivative of $\displaystyle f(x)$ .

$\displaystyle F(x)=\frac{(\ln x)^{2}}{2}$

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Calculate the area bounded by the graph of $\displaystyle f(x)$ , the $\displaystyle x$ -axis and the line $\displaystyle {x}={e}$ .

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The function $\displaystyle {f(x)}={e^x-2x}$ is given.
The drawing shows the graph of $\displaystyle f(x)$ and the graph of the derivative $\displaystyle f'(x)$ .

\displaystyle y=\,?

\displaystyle y=\,?

Which graph is the graph of $\displaystyle f'(x)$ – the first in the list or the second? Explain with properties of the graphs.

Find the $\displaystyle x$ -coordinate of the extremum point of $\displaystyle f(x)$ .

Calculate the $\displaystyle y$ -coordinate of the extremum point.

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

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

Open as a page

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
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$