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Finding Functions Find functions f and g such that
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Calculus
- Limits Let a be a nonzero constant. Prove that if lim x→0f(x) = L then lim x→0f(ax) = L..b. Show by means of an example that a must be nonzero Chapter 2 Differentiation 4. An astronaut standing on the moon throws a rock into the air. The height of the rock iss = −(27/10) t^2 + 27t + 6where s is measured in feet and t is measured in seconds.a. Find expressions for velocity and acceleration of the rock.b. Find the time when the rock is at its highest point. What is the height of the rock at this time?c. Find the total time that the rock spend on the air. Chain Rule a. Show that the derivative of an odd function is even. That is, if f(-x) = -f(x), then f(−x) =-f′(x).b. Show that the derivative of an even function is odd. That is, if f(-x) = f(x), then f(−x) =-f′(x).arrow_forwardDerivatives of even and odd functions Recall that ƒ is evenif ƒ(-x) = ƒ(x), for all x in the domain of ƒ, and ƒ is odd ifƒ(-x) = -ƒ(x), for all x in the domain of ƒ.a. If ƒ is a differentiable, even function on its domain, determinewhether ƒ′ is even, odd, or neither.b. If ƒ is a differentiable, odd function on its domain, determinewhether ƒ′ is even, odd, or neitherarrow_forwardDiscontinuous composite of continuous functions Give an example of functions ƒ and g, both continuous at x = 0, for which the composite ƒ g is discontinuous at x = 0.arrow_forward
- Uniqueness of limits Show that a function cannot have two differentlimits at the same point. That is, if limxSc ƒ(x) = L1 andlimxSc ƒ(x) = L2, then L1 = L2.arrow_forwardlim x approches -infinity x ln(1-1/x)arrow_forwardlim of x approaching infinity of (e^x-3x)^2 / e^2x+ln(x^2) The (e^x-3x)^2 is on the top of the fraction and the e^2x+ln(x^2) is on the bottom of the fractionarrow_forward
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