Problem 32 - Calculus A New Horizon

Question

In each part sketch a continuous curve \(y=f(x)\) with the stated properties. (a) \(f(2)=4, \quad f^{\prime}(2)=0, \quad f^{\prime \prime}(x)<0\) for all \(x\) (b) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)>0\) for \(x<2, f^{\prime \prime}(x)<0\) for \(x>2\) (c) \(f(2)=4, \quad f^{\prime \prime}(x)>0\) for \(x \neq 2\) and \(\lim _{x \rightarrow 2^{+}} f^{\prime}(x)=-\infty$$\lim _{x \rightarrow 2^{-}} f^{\prime}(x)=+\infty\)

Step-by-Step Solution

Answer

Sketch different curves for each part respecting the given conditions: concave down, inflection at 2, and a cusp with upward and downward approach.

Step 1: Understand Property (a)

For this part, we need a curve where at the point \( x = 2 \), \( f(2) = 4 \), \( f'(2) = 0 \), and \( f''(x) < 0 \) for all \( x \). This implies that the curve has a horizontal tangent at \( x = 2 \) and is concave down everywhere.

Step 2: Sketch Curve for (a)

Draw a curve passing through the point \( (2, 4) \) with a horizontal tangent line (since \( f'(2) = 0 \)). Ensure the curve is concave down for all \( x \), like an upside-down parabola centered at (2,4).

Step 3: Understand Property (b)

For part (b), we need \( f(2) = 4 \), \( f'(2) = 0 \), \( f''(x) > 0 \) when \( x < 2 \), and \( f''(x) < 0 \) when \( x > 2 \). This indicates a curve that switches from concave up to concave down at \( x = 2 \), resembling an inflection point.

Step 4: Sketch Curve for (b)

Draw the curve passing through \( (2, 4) \) with a horizontal tangent at \( x = 2 \). To the left of \( x = 2 \), sketch the curve as concave up, and to the right, make it concave down, like a cubic function around the inflection point \( x=2 \).

Step 5: Understand Property (c)

In part (c), we require \( f(2) = 4 \), \( f''(x) > 0 \) for \( x eq 2 \), and a vertical tangent at \( x = 2 \) due to \( \lim_{x \to 2^{+}} f'(x) = -\infty \) and \( \lim_{x \to 2^{-}} f'(x) = +\infty \). This suggests a cusp at \( x = 2 \).

Step 6: Sketch Curve for (c)

Sketch a curve passing through \( (2,4) \) that is concave up on both sides of \( x = 2 \), but approaches a vertical tangent as it nears \( x = 2 \). The cusp at \( x = 2 \) implies the curve slopes drastically upwards from the left and downwards from the right.

Key Concepts

Continuous Curve

In the world of calculus, a continuous curve is a smooth line on a graph that never breaks or lifts its pen off the page. Imagine drawing a line without any interruptions. That’s what a continuous curve looks like. It helps us understand a function's behavior without gaps or jumps, allowing us to predict values in between known points. These curves can model real-world phenomena smoothly, guaranteeing a constant output for a given input.

Concavity

Concavity describes the direction a curve opens or bends. It tells us if the curve is "smiling" or "frowning."
- Concave Up: When a curve looks like a smile. This means the second derivative, denoted as \( f''(x) > 0 \), is positive. The slopes of tangent lines increase as you move right.- Concave Down: The curve resembles a frown, with \( f''(x) < 0 \), meaning the slope of tangent lines decreases as you move to the right.
Understanding concavity helps predict how a function accelerates or decelerates and identifies sections of the graph that are either rising faster or slowing down.

Inflection Point

An inflection point is a special spot on a graph where the curve changes concavity. It’s like a turning point where the curve stops "frowning" and starts "smiling," or vice versa. For example, if a curve goes from concave up to concave down at a certain point, that's an inflection point.
To identify an inflection point, look for when the second derivative changes sign. It’s crucial for understanding changes in the acceleration and deceleration of a function, meaning it’s useful in contexts where growth patterns shift dramatically.

Tangent Line

Imagine a straight line that just touches a curve at a single point without cutting across it. This is a tangent line. It gives you the best linear approximation of the curve at that particular point. The slope of this line is given by the first derivative, \( f'(x) \), indicating how steep the curve is at that point.

• If \( f'(x) = 0 \), the tangent is horizontal, showing a moment where the curve doesn’t rise or fall.
• Positive \( f'(x) \) means an upward tilt, while negative means downward.

Understanding tangent lines is fundamental in calculus because they help solve optimization problems by finding maxima and minima.

Cusp

A cusp occurs on the curve where it suddenly changes direction sharply, looking like a pointed tip.
Unlike smooth transitions in a continuous curve, a cusp features a vertical tangent line where the curve approaches infinity in slope, signifying an abrupt turn.It can be recognized mathematically when the limits of the derivatives from both sides aren't equal, as shown in part (c) of the example: \( \lim_{x \rightarrow c^{+}} f'(x)\) and \( \lim_{x \rightarrow c^{-}} f'(x)\) have different values, such as \(+ \infty\) and \(- \infty\). That's crucial in identifying points on a curve where sharp changes occur, especially in applications involving sudden shifts.