به انگلیسی

Growth that multiplies every step

Mathematics · Algebra · ages 16-17

Name ______________________ Date ____________

An exponential function has outputs 24, 72 and 216 at inputs 0, 1 and 2. What is its multiplier for each unit increase in the input?
Answer: ______________
For a>0 and a≠1, which statement is true of y=a^x?
- It reaches y=0 at a finite input and then stays there.
- It stays positive and has horizontal asymptote y=0.
- It switches between positive and negative outputs forever.
For y=2^(-x), the output approaches zero as x becomes very negative.
Circle one: True False
How does the graph of y=2^(-x) differ from y=2^x?
- It falls to the right, reflected in the y-axis.
- It rises to the right, reflected in the x-axis.
- It rises to the right after an upward shift of two units.
The tangent gradient of y=e^x at x=1 is e, so y increases by exactly e between x=1 and x=2.
Circle one: True False
In N=A e^(kt), with A>0, a negative k means N decays toward zero as elapsed time t increases.
Circle one: True False
A culture doubles each hour from 1 cell. How many cells after 4 hours?
Answer: ______________
For y=2^(-x), find y when x=4. Give your answer as a decimal.
Answer: ______________
An amount starts at 50. Its rate of loss is a fixed fraction of its current size. Which model fits for t≥0?
- N=50e^(kt), with k>0
- N=50+kt, with k<0
- N=50e^(kt), with k<0
N=75e^(kt). Jo sets k=0 and predicts N=0 for every t≥0. Which check corrects Jo's prediction?
- The exponent is 0, so N=75×1=75.
- The exponent is 0, so N=75×0=0.
- The exponent equals t, so N=75e^t and grows over time.
- The starting value is 0, so N=e^t for all later times.

Answer key

For grown-ups. Fold this page away before handing over the rest.

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3 · Divide consecutive outputs: 72/24=3 and 216/72=3. The unit-step multiplier is 3.
It stays positive and has horizontal asymptote y=0. · Every output is positive. The curve approaches y=0 in one direction without reaching it.
False · Very negative x makes -x positive and large, so 2^(-x) grows. This curve approaches zero as x becomes very positive.
It falls to the right, reflected in the y-axis. · Replacing x by -x reflects in the y-axis. The growing curve becomes a decreasing curve.
False · The tangent gradient is e at x=1, but the interval increase is e^2-e=e(e-1). Since e>2, this increase is greater than e.
True · For k<0, the unit-time multiplier e^k is between zero and one. Repeated multiplication makes the positive amount approach zero.
16 · Doubling four times gives 1, 2, 4, 8, 16.
0.0625 · Substitution gives 2^(-4)=1/2^4=1/16=0.0625.
N=50e^(kt), with k<0 · A fixed proportional rate uses the exponential model. The starting amount gives A=50, and loss requires k<0.
The exponent is 0, so N=75×1=75. · Setting k=0 makes the exponent zero, not the output. Since e^0=1, N remains 75.