1. ## calculus with exponentials

simplify: exp(a)^-1 * exp (b)^2/exp(5c)

calculate: d/dx exp(xsin(x))

calculate: (integral)xexp(x^2-3)dx

find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???

find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???

2. Originally Posted by keemariee
simplify: exp(a)^-1 * exp (b)^2/exp(5c)

Iso: You just need to know basic laws of exponentiation. Like $\displaystyle e^x e^y = e^{x+y}$ and $\displaystyle e^{-x} = \frac1{e^x}$

$\displaystyle (e^{a})^{-1} {(e^b)}^2/e^{5c} = e^{-a + 2b -5c}$

calculate: d/dx exp(xsin(x))

Iso: Use chain rule:
$\displaystyle \frac{d(e^{x \sin x})}{dx} = e^{x\sin x} \frac{d(x\sin x)}{dx}=....$

So can you continue now?

calculate: (integral)xexp(x^2-3)dx

Iso: Substitute $\displaystyle u = x^2 - 3$, so that $\displaystyle du = 2x \, dx$

$\displaystyle \boxed{\int x e^{x^2-3}\,dx = \frac12 \int e^{u}\,du = \frac{e^{x^2 - 3}}{2}}$

find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???

find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
..

3. Hi, keemariee! I've rewritten the expressions in your problems, so let me know if I misinterpreted anything.

Originally Posted by keemariee
simplify: $\displaystyle \frac{\text{e}^{-a}\,\text{e}^{2b}}{\text{e}^{5c}}$
Use these properties:

$\displaystyle a^ma^n = a^{m+n}$

$\displaystyle \frac{a^m}{a^n} = a^{m-n}$

Originally Posted by keemariee
calculate: $\displaystyle \frac{d}{dx} \left[\text{e}^{x\sin x}\right]$
Use the chain rule:

$\displaystyle \frac{d}{dx} \left[\text{e}^{x\sin x}\right]$

$\displaystyle =\text{e}^{x\sin x}\frac{d}{dx}\left[x\sin x\right]$

And now apply the product rule.

Originally Posted by keemariee
calculate: $\displaystyle \int x\,\text{e}^{x^2-3}\,dx$
Use substitution if you have to (letting $\displaystyle u = x^2 - 3$), or just bring in a factor of 2 and recognize that

$\displaystyle 2x\,\text{e}^{x^2 - 3} = \text{e}^{x^2 - 3}\frac{d}{dx}\left[\text{e}^{x^2 - 3}\right]$

You can apply the chain rule in reverse.

Originally Posted by keemariee
find the exponential function $\displaystyle f(x)=a^x$ whose graph goes through point $\displaystyle \left(5,\ \frac1{32}\right)$. $\displaystyle a=\text{???}$
Substitute the values:

$\displaystyle f(5) = \frac1{32}\Rightarrow a^5 = \frac1{32}$

Now solve for $\displaystyle a$:

$\displaystyle a = \sqrt[5]{\frac1{32}} = \frac1{\sqrt[5]{32}}$

What value can you raise to the 5th power to get 32?

Originally Posted by keemariee
find the exponential funtion $\displaystyle f(x)=Ca^x$ whose graph goes through points $\displaystyle (0,\ 2)$ and $\displaystyle (4,\ 32)$. $\displaystyle a=\text{???},\ C=\text{???}$
Do the same thing as the last problem. The only difference is that you will have two equations with two unknowns, so you will need to solve a system.

4. Hello,

Originally Posted by keemariee
find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???
We know it goes through (5,1/32)

--> $\displaystyle f(5)=a^{5}=\frac{1}{32}$

But $\displaystyle 32=2^5 \implies a=\frac 12$

find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
Again :
$\displaystyle 2=f(0)=Ca^0 \implies \boxed{C=2}$

$\displaystyle f(4)=32 \implies 32=2 \cdot a^4$

$\displaystyle a^4=16$

But $\displaystyle 16=2^4 \implies a=2$

5. Originally Posted by keemariee
find the exponential function f(x)=a^x whose graph goes through point 5, 1/32. a=???
$\displaystyle \frac1{32} = f(5) = a^5 \Rightarrow a^5 = \frac1{32} = \left(\frac12 \right)^5 \Rightarrow a = \frac12$

Originally Posted by keemariee
find the exponential funtion f(x)=Ca^x whose graph goes through points (0,2) and (4,32). a=??? C=???
$\displaystyle 2 = f(0) = Ca^0 = C$

$\displaystyle 32 = f(4) = Ca^4 \Rightarrow 2a^4 = 32 \Rightarrow a^4 = 16 \Rightarrow a = \pm2$

So actually there are two graphs...
$\displaystyle f(x) = 2(-2)^x$ and $\displaystyle f(x) = 2^{x+1}$