34 Function Problems : Graphing, Translation, Reflection, Domains, Limits, Derivatives and Tangents

For the following graph the given functions on a computer screen, how are these graphs related?

1) Y=2^x, y=e^x, y=5^x, y=20^x

2) Y=3^x, y=10^x, y=(1/3)^x, y=(1/10)^x

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Make a sketch of the function.

7) y=4^x-3
8) y=-2x^-x
9) y=3-e^x

13) Starting with the graph of y=e^x, write the equation of the graph that results from.
a) shifting two units downwards.
b) shifting two units to the right.
c) reflecting about the x axis.
d) reflecting about the y axis.
e) ) reflecting about the x axis and then about the y axis.

Find the domain of the functions:

a) f(x)=1/(1+e^x)
b) f(x) = 1/(1-e^x)
c) g(t)=sin(e^-1)
d) g(t) = SQRT(1-2^t)

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19) Suppose the graphs of f(x)=x^2 and g(x)=2^x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that at a distance of 2 feet to the right of the origin, the height of the graph of f is 48ft but the height of the graph of g is about 265mi.
20) Compare the functions f(x)=x^10 and g(x)=e^x by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f?

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Find the limits for the following

1) lim as x approaches infinity(1.001)^x
2) lim as x approaches infinity(e)^-2x
3) lim as x approaches infinity(e^3x - e^-3x)/(e^3x + e^-3x)
4) lim as x approaches (pi/2)^+ (e^tanx)
5) lim as x approaches 2^+ (e^(3/(2-x)))
6) lim as x approaches 2^- (e^(3/(2-x)))

Differentiate the functions:

1) f(x) =x^2e^2
2) y=e^ax^3
3) f(u) =e^(1/u)
4) f(t)=e^(tsin2t)
5) y=SQRT(1+2e^3x)
6) y=e^e^x
7) y= (ae^x+b)/(ce^x+d)
8) y=e^x/(1+x)
9) g(x)=SQRT(x)e^x
10) y=e^(ktanSQRT(x))
11) y=SQRT(1+xe^-2x)

Find the equations of the tangent line to the curve at the given point.

1) y=e^(2x) cos(pi)(x), (0,1)