2) te^-t + 5
I assume you're trying to sketch these functions based on information on f(x) (intercepts, domain, asymptotes) ; f'(x) (critical numbers), and possibly f''(x).
Start by taking the derivative in each case, and factor.
For example, problem 2:
The domain is all real numbers.
The y intercept is 5
y=5 is a horizontal asymptote since at t goes to infinity, t / e^t goes to zero.
The first derivative:
We use the factored form of the first derivative to decide over which intervals the function increases or decreases:
e^t is always positive, so the function will be decreasing whenever t > 1 and increasing when t < 1. Note that because it is continuous, this makes t=1 a local max.
The second derivative:
When t < 2, the function f(t) is concave up because f"(t) is positive.
When t > 2, the function f(t) is concave down because f"(t) is negative.
Use all the information above to sketch this function by hand.
Follow similar steps for your other problem and all problems like these.
If you're having trouble graphing these with a graphing utility or device, let us know.