Chapter 2
Models of Growth: Rates of Change
2.4
Exponential Functions
Exercises
- Suppose you invest \(\$\)4000 at an annual interest rate of 5.25% compounded annually. How much would you have in 20 years?
- How much would you have to invest at 4.75% compounded annually so that you would have \(\$\)20,000 after 10 years?
- How much would you have to invest at 5.15% compounded annually so that you would have \(\$\)30,000 after 20 years?
- Rewrite each of the following equations in exponential form.
a. \(\log_9 3=1/2\) b. \(\log_{10} 1000=3\) c. \(\log_{10} 0.1=-1\) d. \(\log_2 4=2\) e. \(\log_{10} 100=2\) f. \(\log_{10} 0.01=-2\) -
Find the value of each of the following expressions.
a. \(\ln 1\) b. \(\ln e\) c. \(\ln e^3\) d. \(\ln e^{2.7183}\) e. \(\ln 1/e\) f. \(\ln 1/e^2\) -
Solve each of the following equations for \(t\).
a. \(t^6=10\) b. \(e^t=10\) c. \(10^t=6\) -
Calculate each of the following derivatives.
a. \(d/dt e^{-2t}\) b. \(d/dt e^{0.07t}\) c. \(d/dt 2^t\) d. \(d/dt (2t-5e^t)\) e. \(d/dt 4t^3\) f. \(d/dt e\) - Express \(2e^{3t}\) in the form \(c b^t\).
- Express \(15e^{0.15t}\) in the form \(c b^t\).
- Express \(2*3^t\) in the form \(c e^{kt}\).
- Express \(3*10^t\) in the form \(c e^{kt}\).
- Find the derivative of each of the following functions.
a. \(t^4-2t^3+2t^2-t-1\) b. \(t^5+t^3-t^2-9+e^{-t/3}\) c. \(4 e^4-3 e^3+2 e^2-e+7\) d. \(13-26t +6t^2+e^t\) e. \(2 e^{3t}-3t^2\) f. \(1/e^{2t}\) g. \(t^{10}+10^t\) h. \(2/10^t\)
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Find the slope of the graph of \(y=e^t\) at each of the following points.
a. \((2,e^2)\) b. \((-2,e^{-2})\) -
Find the slope of the graph of \(y=2^t\) at each of the following points.
a. \((2,4)\) b. \((-2,0.25)\) -
Find the slope of the graph of \(y=10^t\) at each of the following points.
a. \((2,100)\) b. \((-2,0.01)\) -
Find the slope of the graph of \(y=3^t\) at each of the following points.
a. \((2,9)\) b. \((-2,1/9)\)
