Precalculus Exponential Functions and the Number e Worksheet Solutions (PDF)
This exponential practice solutions for Precalculus covers exponential functions and the number e (Common Core F-LE.A.2, F-IF.C.8b, F-LE.B.5, F-IF.C.7e), with a free PDF to print.
Precalculus · Common Core State Standards: F-LE.A.2, F-IF.C.8b, F-LE.B.5, F-IF.C.7e
Exponential practice solutions: exercises 1 to 10
Exercise 1 – Growth or decay?
(a) \(b=1.08>1\): growth, \(b=1+0.08\). (b) \(b=0.85< 1\): decay, \(b=1-0.15\).
(a) growth, 8%; (b) decay, 15%
Exercise 2 – Evaluate
\(f(0)=3\), \(f(1)=6\), \(f(4)=3(16)\).
\(3,\ 6,\ 48\)
Exercise 3 – Equation from two points
\(a=f(0)=5\). \(b=\frac{15}{5}=3\).
\(y=5\cdot3^{x}\)
Exercise 4 – Equation from points two apart
Dividing: \(b^{2}=\frac{24}{6}=4\), so \(b=2\). Then \(a\cdot2=6\), \(a=3\).
\(y=3\cdot2^{x}\)
Exercise 5 – Town population
\(P(t)=4000(1.03)^{t}\). \(P(10)=4000(1.3439)\approx 5{,}376\).
\(P(t)=4000(1.03)^{t}\); about \(5{,}376\) people
Exercise 6 – Half-life
\(A(t)=80\left(\frac12\right)^{t/5}\). At \(t=15\): \(80\left(\frac12\right)^{3}=10\).
\(10\text{ mg}\)
Exercise 7 – Transformed exponential
The graph is \(2^{x}\) shifted right 1 and up 3. Asymptote \(y=3\). The \(y\)-intercept: \(2^{-1}+3=3.5\).
\(y=3\); domain all reals; range \((3,\infty)\); \(y\)-int \(3.5\)
Exercise 8 – Equal bases
\(32=2^{5}\), so the exponents must match.
\(x=5\)
Exercise 9 – Quarterly compounding
\(A=2000\left(1+\frac{0.04}{4}\right)^{4\cdot6}=2000(1.01)^{24}\approx \$2{,}539.47\).
\(\$2{,}539.47\)
Exercise 10 – Compare compounding
Yearly: \(1000(1.05)^{10}\approx \$1{,}628.89\). Monthly: \(1000\left(1+\frac{0.05}{12}\right)^{120}\approx \$1{,}647.01\). Continuous: \(1000e^{0.5}\approx \$1{,}648.72\).
\(\$1{,}628.89\); \(\$1{,}647.01\); \(\$1{,}648.72\)
Exercises 11 and up
Exercise 11 – Continuous growth
\(A=500e^{0.035\cdot 8}=500e^{0.28}\approx \$661.56\).
\(\$661.56\)
Exercise 12 – Approach e
\((1.01)^{100}\approx 2.7048\), which is about \(0.0135\) below \(e\).
\(2.7048\), close to \(e\) but still smaller
Exercise 13 – Bacteria
\(N(12)=500\cdot3^{3}=13{,}500\). For \(1500\): \(3^{t/4}=3\), so \(\frac t4=1\).
\(13{,}500\) bacteria; \(t=4\) hours
Exercise 14 – Car depreciation
\(V(t)=22000(0.88)^{t}\). \(V(3)=22000(0.681472)\approx \$14{,}992.38\).
\(V(t)=22000(0.88)^{t}\); \(\$14{,}992.38\)
Exercise 15 – Equation from a graph
\(a=3\); \(\frac{6}{3}=\frac{12}{6}=2\). So \(f(x)=3\cdot2^{x}\) and \(f(5)=3\cdot32=96\).
\(f(x)=3\cdot2^{x}\); \(f(5)=96\)
Exercise 16 – Linear versus exponential
\(f(1)=150< 160\). \(f(2)=225>220\). \(f(3)=337.5>280\). \(f(4)=506.25>340\).
\(f\) overtakes \(g\) at \(x=2\) and stays above
Exercise 17 – Constant percent growth
\(1200b^{2}=1728\), so \(b^{2}=1.44\) and \(b=1.2\).
\(20\%\) per year
Exercise 18 – Find the principal
\(P(1.06)^{3}=1191.02\), so \(P=\frac{1191.02}{1.191016}\approx \$1{,}000.00\).
\(\$1{,}000\)
Exercise 19 – Doubling time
\(1.07^{10}\approx1.967< 2\). \(1.07^{11}\approx2.105\ge 2\).
11 years
Exercise 20 – Cooling coffee
\(T(0)=70+110=180\). \(T(5)=70+110(0.59049)\approx 135.0\). As \(t\) grows, \(0.9^{t}\to0\), so \(T\to70\) (the room temperature).
\(180^{\circ}\text{F}\); about \(135.0^{\circ}\text{F}\); \(70^{\circ}\text{F}\)
Key facts behind this exponential practice solutions
- f(x) = a b^x: a = f(0) and b = growth or decay factor.
- A rate r per period gives b = 1 + r (growth) or b = 1 – r (decay).
- The parent graph has the horizontal asymptote y = 0; y = a b^(x – h) + k has asymptote y = k.
- Compound interest: A = P(1 + r/n)^(nt); continuous: A = Pe^(rt).
- e is about 2.71828, the limit of (1 + 1/n)^n.
How to use this exponential practice solutions
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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