When is y=a^x increasing and when decreasing?

Increasing for a1; decreasing for 0<a<1

Why is the exponential a good model of growth?

For y=Ae^{kx}, \frac{dy}{dx}=ky: the rate of change is proportional to the quantity

Exponential functionsAQA A-Level Maths: Flashcards

Card 1 of 130 of 13 known

Question

What point does every curve $y=a^x$ pass through?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

What point does every curve $y=a^x$ pass through?: $(0,1)$ , because $a^0=1$
What is the asymptote of $y=a^x$ ?: The $x$ -axis, $y=0$
When is $y=a^x$ increasing and when decreasing?: Increasing for $a>1$ ; decreasing for $0<a<1$
$\left(\frac12\right)^x$ written with base $2$ ?: $2^{-x}$
What is special about the gradient of $e^x$ ?: It equals $e^x$ at every point
What is $\frac{d}{dx}\left(e^{kx}\right)$ ?: $ke^{kx}$
Differentiate $5e^{-2x}$ .: $-10e^{-2x}$
Differentiate $20+70e^{-0.05t}$ .: $-3.5e^{-0.05t}$
Why is the exponential a good model of growth?: For $y=Ae^{kx}$ , $\frac{dy}{dx}=ky$ : the rate of change is proportional to the quantity
What does $k<0$ in $y=Ae^{kx}$ model?: Exponential decay
Value of $e^0$ ?: $1$
What happens to $e^{-kt}$ as $t\to\infty$ (for $k>0$ )?: It tends to $0$
Gradient of $y=e^{2x}$ at $x=\frac12$ ?: $2e^{1}=2e$

Exam questions on Exponential functions

The curve $C$ has equation $y=2^x$ .
Find the $x$ -coordinate of each point where $C$ meets the lines $y=16$ and $y=\frac18$ .2 marks
The number $N$ of bacteria in a culture, $t$ hours after it is first measured, is modelled by $N=500e^{0.4t}$ .
Find the rate of increase of the number of bacteria when $t=5$ , giving your answer to 3 significant figures.2 marks
The curve $C$ has equation $y=e^{2x}$ and passes through the point $P$ where $x=\frac12$ .
Find the exact coordinates of $P$ and the exact gradient of $C$ at $P$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).