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MATH 105 Final Exam Review I
- Consider the function f(x) = (^5) −^3 2 x.
(a) Is this function continuous on the domain (−∞, ∞)? Explain.
(b) Compute the average rate of change of f on [1.5, 2].
(c) Using the limit definition of the derivative, compute f′(x).
(d) Find the equation of the tangent line to f at x = 2.
- Given that f(0) = 2, g(0) = 3, f′(0) = 5, g′(0) = 7, and f′^ (3) = π compute the following. (a) h′(0) if h(x) = f(x)g(x)
(b) j′(0) if j(x) = f g((xx))
(c) k′(0) if k(x) = f(g(x))
- Sketch a graph of a function which is always positive and decreasing and which satisfies the following:
x→−∞lim f(x) =^ ∞;^ xlim→∞ f(x) = 2;^ xlim→ 1 −^ f(x) = 5;^ xlim→ 1 +^ f(x) = 4
- Compute dy/dx for each of the following.
(a) y = x^2 + 2x^ + e^2 + x 2 +^2 x + ln (2x) + arctan (2x) + ln(2) + sin 2
(b) y = √x cos(7x^3 )
(c) y = e x (^) + π tan 4 − 7 x
(d) y = tan (ex^2 arcsin(5x))
(e) y^3 + yx^2 + x^2 = 3y^2
- Consider the differential equation y′^ = − 3 y. (a) Sketch the slope field for this DE.
(b) Verify that y = Ce−^3 x^ is a solution for all values of C.
(c) Find the solution that passes through (1, 5).
- Find all possible antiderivatives of the following. (a) g′(t) = e^5 + t^5 + e^5 t
(b) h′(r) = 3 sin(2r) + √^3 r
- Evaluate the following limits.
(a) (^) xlim→∞^ x 2 ln x
(b) (^) xlim→ 0 sin (12x x 3 )^ −^12 x
(c) (^) xlim→ 0 e x (^) − 1 cos x
(d) (^) xlim→ 5352 x^ −−^710 x
(e) (^) xlim→ 0 + x^3 ln x
(f) (^) xlim→ 0 −^1 x
(g) (^) xlim→ 0 x^1
