How do you find the exponential model #y=ae^(bx)# that goes through the points (0,4) and (5, 1/2)?
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To find the exponential model ( y = ae^{bx} ) that goes through the points (0,4) and (5, 1/2), you need to solve for the constants ( a ) and ( b ) using these points.

Plug in the coordinates of the first point (0,4) into the equation: [ y = ae^{bx} ] This gives you: [ 4 = ae^{b \times 0} ] [ 4 = a \times e^0 ] [ 4 = a \times 1 ] [ a = 4 ]

Now, plug in the coordinates of the second point (5, 1/2) into the equation, using the value of ( a ) you found in the previous step: [ \frac{1}{2} = 4 \times e^{b \times 5} ] [ \frac{1}{8} = e^{5b} ]

Take the natural logarithm of both sides to solve for ( b ): [ \ln\left(\frac{1}{8}\right) = \ln(e^{5b}) ] [ \ln\left(\frac{1}{8}\right) = 5b ] [ b = \frac{\ln\left(\frac{1}{8}\right)}{5} ]
Thus, you have found the values of ( a ) and ( b ). Plug these values back into the exponential model equation to get the final equation.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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