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MATH 105 Exam 1 Review February 9, 2005 Basic Derivatives Basic Antiderivatives d dx (constant)= d dx xn^ = antiderivative of xn^ = if n 6 = − 1 d dx ex^ = antiderivative of ex^ = d dx bx^ = antiderivative of bx^ = d dx logb^ x^ = d dx ln x = antiderivative of
x
d dx sin x = antiderivative of sin x = d dx cos x = antiderivative of cos x = d dx [kf (x)] = antiderivative of kf (x) = d dx [f (x) + g(x)] = antiderivative of [f (x) + g(x)] =
Limit Definition of Derivative
f ′(x) =
Graphical Relationships Between f , f ′, and f ′′
f positive negative increasing decreasing concave up concave down f ′ f ′′
Exponent Rules
Logarithm Rules
- logb(xy) =
- logb(x/y) =
- logb(xy^ ) =
- Let f (x) = x + 1 x^2 − 1
(a) What is the natural domain of f? (b) Where if f discontinuous? (c) What is lim x→ 3 f (x)? (d) What is (^) xlim→− 1 f (x)? (e) What is lim x→ 1 f (x)? (f) What is (^) xlim→1+ f (x)?
- Given the data in the table below, estimate f ′(6). x 5.97 6.00 6. f (x) 11.55 11.04 10. If you had to make a guess at the sign of f ′′(6), what would it be?
- Let f (x) = 6 8 − 3 x
(a) Compute the average rate of change of f on the interval [3,6].
(b) Using the limit definition of the derivative, find f ′(x).
(c) Find the equation of the tangent line to f at x = 4.
(d) How would the derivative of g(x) = f (x) + 5 compare to f ′(x)?
(e) How would the derivative of h(x) = f (x + 5) compare to f ′(x)?
- Suppose that T (t) gives the temperature in Lewiston as a function of time. In each of the following situations, determine if the signs of T , T ′, and T ′′^ are positive, negative, zero, or unknown. (a) The temperature is 60 degrees and falling steadily.
(b) The temperature is rising more and more slowly.
(c) The temperature is -5 degrees and rising.
- Solve the differential equation y′(x) = − 4 x + 10 if y(1) = 3.
- An object has vertical velocity v(t) = t^2 − 5 t + 4 feet per second on the interval [0, 5]. A positive velocity indicates the object is ascending, and a negative velocity indicates it is descending. At time t = 0, the object is 50 feet above ground. (a) When is the object ascending? When is it descending?
(b) When is the object’s acceleration most positive?
(c) What is the greatest height the object reaches?
- Find the derivatives of the following.
(a) y = 2 + 3x^ + x^4 + x 5
x
x^8
x + π^10 + sin(11) + log 12 x + 13 cos x
(b) y = ln(x^2 e^3 x)
(c) y = 23+4x
- Find antiderivatives of the following.
(a) y = π + ex^ − 2(3x)
(b) y = x^4 − 5 sin x +
x
