The function $$ y(t) $$ is given by $$ y=\mathrm{e}^{t} $$ (a) Calculate values of $$ t $$ such that $$ y $$ is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if $$ t $$ is increased by $$ \ln 2 $$ then the value of $$ y $$ doubles. (c) A function $$ z(t) $$ is given by $$ z=\mathrm{e}^{t} \quad k0 $$ Find the increase in the value of $$ t $$ which will result in the value of $$ z $$ doublin

Question

The function $$ y(t) $$ is given by $$ y=\mathrm{e}^{t} $$ (a) Calculate values of $$ t $$ such that $$ y $$ is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if $$ t $$ is increased by $$ \ln 2 $$ then the value of $$ y $$ doubles. (c) A function $$ z(t) $$ is given by $$ z=\mathrm{e}^{t} \quad k>0 $$ Find the increase in the value of $$ t $$ which will result in the value of $$ z $$ doublin

UpStudy AI · Accepted Answer

To solve the problem, we will go through each part step by step.

### Part (a)
We need to find values of $$ t $$ such that $$ y(t) = \mathrm{e}^{t} $$ equals specific values.

1. **For $$ y = 1 $$**:
   $$
   \mathrm{e}^{t} = 1
   $$
   Taking the natural logarithm of both sides:
   $$
   t = \ln(1) = 0
   $$

2. **For $$ y = 2 $$**:
   $$
   \mathrm{e}^{t} = 2
   $$
   Taking the natural logarithm:
   $$
   t = \ln(2)
   $$

3. **For $$ y = 4 $$**:
   $$
   \mathrm{e}^{t} = 4
   $$
   Taking the natural logarithm:
   $$
   t = \ln(4) = \ln(2^2) = 2\ln(2)
   $$

4. **For $$ y = 8 $$**:
   $$
   \mathrm{e}^{t} = 8
   $$
   Taking the natural logarithm:
   $$
   t = \ln(8) = \ln(2^3) = 3\ln(2)
   $$

### Summary of Part (a):
- $$ t $$ for $$ y = 1 $$: $$ t = 0 $$
- $$ t $$ for $$ y = 2 $$: $$ t = \ln(2) $$
- $$ t $$ for $$ y = 4 $$: $$ t = 2\ln(2) $$
- $$ t $$ for $$ y = 8 $$: $$ t = 3\ln(2) $$

### Part (b)
We need to show that if $$ t $$ is increased by $$ \ln 2 $$, then the value of $$ y $$ doubles.

Let $$ t' = t + \ln(2) $$. Then:
$$
y(t') = \mathrm{e}^{t'} = \mathrm{e}^{t + \ln(2)} = \mathrm{e}^{t} \cdot \mathrm{e}^{\ln(2)} = \mathrm{e}^{t} \cdot 2
$$
Thus, we have:
$$
y(t') = 2y(t)
$$
This shows that increasing $$ t $$ by $$ \ln(2) $$ results in $$ y $$ doubling.

### Part (c)
Now we consider the function $$ z(t) = \mathrm{e}^{t} $$ and we want to find the increase in $$ t $$ that will result in $$ z $$ doubling.

Let $$ t' = t + \Delta t $$. We want:
$$
z(t') = 2z(t)
$$
This means:
$$
\mathrm{e}^{t + \Delta t} = 2\mathrm{e}^{t}
$$
Dividing both sides by $$ \mathrm{e}^{t} $$:
$$
\mathrm{e}^{\Delta t} = 2
$$
Taking the natural logarithm of both sides:
$$
\Delta t = \ln(2)
$$

### Summary of Part (c):
The increase in the value of $$ t $$ which will result in the value of $$ z $$ doubling is $$ \Delta t = \ln(2) $$.
- Part (a): $$ t $$ for $$ y = 1 $$: $$ t = 0 $$, for $$ y = 2 $$: $$ t = \ln(2) $$, for $$ y = 4 $$: $$ t = 2\ln(2) $$, for $$ y = 8 $$: $$ t = 3\ln(2) $$. - Part (b): Increasing $$ t $$ by $$ \ln(2) $$ doubles $$ y $$. - Part (c): The increase in $$ t $$ to double $$ z $$ is $$ \Delta t = \ln(2) $$.

The function is given by

(a) Calculate values of such that is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 .
(b) Show that if is increased by then the value of doubles.
© A function is given by

Find the increase in the value of which will result in the value of doublin

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Answer

Solution

Part (a)

Summary of Part (a):

Part (b)

Part ©

Summary of Part ©:

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The function is given by (a) Calculate values of such that is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if is increased by then the value of doubles. © A function is given by Find the increase in the value of which will result in the value of doublin